MX Algebra Solve Quadratic Equations Using the Quadratic Formula

Section 9.3Solve Quadratic Equations Using the Quadratic Formula

Definition

Before you get started, take this readiness quiz.

1

Evaluate \({b}^{2}-4ab\) when \(a=3\) and \(b=-2.\)
If you missed this problem, review Example 10.

\(28\)

Definition
2

Simplify: \(\sqrt{108}.\)
If you missed this problem, review Example 1.

\(6\sqrt{3}\)

Definition
3

Simplify: \(\sqrt{50}.\)
If you missed this problem, review Example 1.

\(5\sqrt{2}\)

Solve Quadratic Equations Using the Quadratic Formula

When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.

We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for x.

We start with the standard form of a quadratic equation and solve it for x by completing the square.

Table 1
A quadratic equation, ax^2 + bx + c = 0, where 'a' cannot be zero, is shown in this image. It's a fundamental algebraic expression used to find the roots of a second-degree polynomial.
Isolate the variable terms on one side.A mathematical equation displays 'ax^2 + bx = -c' in gray text against a plain white background, representing a quadratic equation.
Make the coefficient of \({x}^{2}\) equal to 1, by
dividing by a.		A mathematical equation showing the division of a quadratic equation by 'a', resulting in x^2 + (b/a)x = -c/a, a step in solving for x by completing the square.
Simplify.A mathematical equation is displayed, showing x squared plus b over a times x equals negative c over a, representing a step in solving a quadratic equation by completing the square.
To complete the square, find \({(\frac{1}{2}·\frac{b}{a})}^{2}\) and add it to both sides of the equation.
\({(\frac{1}{2}\frac{b}{a})}^{2}=\frac{{b}^{2}}{4{a}^{2}}\)A mathematical equation demonstrating a step in completing the square: x^2 + (b/a)x + (b^2)/(4a^2) = -c/a + (b^2)/(4a^2), with the term b^2/(4a^2) highlighted in red on both sides.
The left side is a perfect square, factor it.The mathematical equation (x + b/(2a))^2 = -c/a + b^2/(4a^2) is shown on a white background. It represents an intermediate step in solving quadratic equations.
Find the common denominator of the right
side and write equivalent fractions with
the common denominator.		A mathematical equation illustrating a step in completing the square, showing (x + b/(2a))^2 equals b^2/(4a^2) minus (c/a) multiplied by (4a/4a).
Simplify.A mathematical equation illustrating a step in solving for x using the completing the square method, which is commonly used to derive the quadratic formula.
Combine to one fraction.A mathematical equation shows (x + b/(2a))^2 = (b^2 - 4ac)/(4a^2), representing an intermediate step in deriving the quadratic formula by completing the square.
Use the square root property.A mathematical equation shows x plus b over 2a equals plus or minus the square root of b squared minus 4ac, all divided by 4a squared, illustrating a step in deriving the quadratic formula.
Simplify the radical.A mathematical equation shown as x + b/(2a) = +/- sqrt(b^2 - 4ac) / (2a), a step in deriving the quadratic formula, isolated on a white background.
Add \(-\frac{b}{2a}\) to both sides of the equation.The quadratic formula, showing the solutions for x are equal to negative b divided by two a, plus or minus the square root of b squared minus four a c, all divided by two a.
Combine the terms on the right side.The quadratic formula, x = (-b ×1×4b^2 - 4ac) / 2a, is displayed, used to find the roots of a quadratic equation. This fundamental algebraic equation is a key tool in mathematics for solving polynomials of degree 2.
This equation is the Quadratic Formula.
Quadratic Formula

The solutions to a quadratic equation of the form ax2 + bx + c = 0, where \(a\ne 0\) are given by the formula:

\[x=\frac{\text{-}b\pm \sqrt{{b}^{2}-4ac}}{2a}\]

To use the Quadratic Formula, we substitute the values of a, b, and c from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.

Notice the formula is an equation. Make sure you use both sides of the equation.

Example 1How to Solve a Quadratic Equation Using the Quadratic Formula

Solve by using the Quadratic Formula: \(2{x}^{2}+9x-5=0.\)

Identify a, b, and c from the standard form, then substitute them into the Quadratic Formula.

Step 1 is to write the quadratic equation in standard form, a times x squared plus b x plus c equals zero, and identify the values a, b, and c. The equation 2 x squared plus 9 x minus 5 equals zero is in standard form. A equals 2, b equals 9, and c equals negative 5. Step 2. Write the quadratic formula. Then substitute the values of a, b, and c. Substitute a equals 2, b equals 9, and c equals negative 5 into the equation x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. So x equals the quotient negative 9 plus or minus the square root of the difference 9 squared minus the product 4 times 2 times negative 5 divided by the product 2 times 2. In step 3, simplify the fraction and solve for x. x equals the quotient negative 9 plus or minus the square root of the difference 81 minus negative 40 divided by 4. Simplify the radicand. x equals the quotient negative 9 plus or minus the square root of 121 divided by 4. Simplify the square root. x equals the quotient negative 9 plus or minus 11 divided by 4. Separate into two equations. The first equation is x equals the quotient negative 9 plus 11 divided by 4 which simplifies to 2 divided by 4. The first solution is x equals one half. The second equation is x equals the quotient negative 9 minus 11 divided by 4 which simplifies to negative 20 divided by 4. The second solution is x equals negative 5. The fourth, and final, step is to check the solution. Put each answer into the original equation to check. First, substitute x equals one half into the original equation, 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of one half plus nine times one half minus 5. We need to show that this expression equals 0. Simplify the square. 2 times one fourth plus nine times one half minus 5 equals one half plus 9 halves minus 5, or 10 halves minus 5. 5 minus 5 equals 0, so x equals one half is indeed a solution. Next substitute x = negative 5 into the equation 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of negative 5 plus 9 times negative 5 minus 5. We need to show that this expression equals 0. Simplify the square. 2 times 25 plus nine times negative 5 minus 5 equals 50 minus 45 minus 5, or 0. x equals negative 5 is a solution as well.
Try It #1

Solve by using the Quadratic Formula: \(3{y}^{2}-5y+2=0\) .

\(y=1,y=\frac{2}{3}\)

Did you get it?
Try It #2

Solve by using the Quadratic Formula: \(4{z}^{2}+2z-6=0\) .

\(z=1,z=-\frac{3}{2}\)

Did you get it?
Solve a quadratic equation using the quadratic formula.
  • Write the quadratic equation in standard form, ax2 + bx + c = 0. Identify the values of a, b, and c.
  • Write the Quadratic Formula. Then substitute in the values of a, b, and c.
  • Simplify.
  • Check the solutions.

If you say the formula as you write it in each problem, you’ll have it memorized in no time! And remember, the Quadratic Formula is an EQUATION. Be sure you start with “x =”.

Example 2

Solve by using the Quadratic Formula: \({x}^{2}-6x=-5.\)

Rewrite the equation in standard form first, then identify a, b, and c.

Table 2
A mathematical equation is displayed, showing 'x squared minus 6x equals minus 5' against a white background.
Write the equation in standard form by adding
5 to each side.		A clear, focused image displays the quadratic equation x^2 - 6x + 5 = 0 presented in black text against a plain white background.
This equation is now in standard form.Two quadratic equations are displayed: the general form ax^2 + bx + c = 0 in red, and a specific example x^2 - 6x + 5 = 0 in black.
Identify the values of \(a,\,\,b,\,\,c.\)The image displays mathematical coefficients a=1 in light blue, b=-6 in red, and c=5 in light green, set against a plain white background.
Write the Quadratic Formula.The quadratic formula, used to find the solutions for x in a quadratic equation of the form ax^2 + bx + c = 0.
Then substitute in the values of \(a,\,\,b,\,\,c.\)The quadratic formula, x = -(-6) ×177 sqrt((-6)^2 - 4 * 1 * 5) / (2 * 1), demonstrates the substitution of values for 'a', 'b', and 'c' to solve a quadratic equation.
Simplify.

A quadratic formula step showing x equals six plus or minus the square root of thirty-six minus twenty, all divided by two. A mathematical equation shows x equals the fraction of (6 plus or minus the square root of 16) all over 2. This is a step in solving a quadratic equation. A mathematical equation displays x equals 6 plus or minus 4, all divided by 2.
Rewrite to show two solutions.Two mathematical expressions for 'x' are displayed: x = (6+4)/2 and x = (6-4)/2, representing two possible solutions in a mathematical context.
Simplify.Two mathematical equations are displayed horizontally: x = 10/2, and x = 2/2.
The image shows two mathematical expressions on a white background: 'x = 5,' and 'x = 1'.
Check:

Verification of solutions for the quadratic equation x^2 - 6x + 5 = 0, demonstrating that both x=5 and x=1 satisfy the equation, resulting in 0=0.
Try It #3

Solve by using the Quadratic Formula: \({a}^{2}-2a=15\) .

\(a=-3,a=5\)

Did you get it?
Try It #4

Solve by using the Quadratic Formula: \({b}^{2}+24=-10b\) .

\(b=-6,b=-4\)

Did you get it?

When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the Quadratic Formula. If we get a radical as a solution, the final answer must have the radical in its simplified form.

Example 3

Solve by using the Quadratic Formula: \(2{x}^{2}+10x+11=0.\)

Substitute a, b, and c into the formula and simplify the radicand; expect an irrational result.

Table 3
A quadratic equation is displayed, reading '2x^2 + 10x + 11 = 0'.
This equation is in standard form.The general form of a quadratic equation, ax^2 + bx + c = 0, is shown above a specific example: 2x^2 + 10x + 11 = 0.
Identify the values of a, b, and c.The image displays the values of three variables: a = 2, b = 10, and c = 11, written in a horizontal line with different colors for each variable and its value.
Write the Quadratic Formula.The image displays the quadratic formula, an equation used to solve quadratic equations for the variable x. It shows x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.
Then substitute in the values of a, b, and c.The quadratic formula is shown with values substituted for solving an equation: x = (- (10) +/- sqrt((10)^2 - 4 * 2 * (11))) / (2 * 2). The numbers 10, 2, and 11 are color-coded in red, cyan, and yellow, respectively.
Simplify.A quadratic formula calculation shows x equals negative 10 plus or minus the square root of 100 minus 88, all divided by 4.
A mathematical equation showing x equals the fraction with a numerator of -10 plus or minus the square root of 12, all divided by 4. This is a common step in the quadratic formula.
Simplify the radical.The equation shows x equals the fraction with numerator -10 plus or minus 2 times the square root of 3, and denominator 4.
Factor out the common factor in the numerator.The equation x equals the fraction 2 multiplied by the quantity -5 plus or minus the square root of 3, all divided by 4, is displayed on a white background.
Remove the common factors.Equation showing x equals negative five plus or minus the square root of three, all divided by two.
Rewrite to show two solutions.Two mathematical expressions for x are displayed. The first is x = (-5 + sqrt(3))/2, and the second is x = (-5 - sqrt(3))/2, representing solutions to a quadratic equation.
Check:
We leave the check for you!
Try It #5

Solve by using the Quadratic Formula: \(3{m}^{2}+12m+7=0\) .

\(m=\frac{-6+\sqrt{15}}{3},m=\frac{-6-\sqrt{15}}{3}\)

Did you get it?
Try It #6

Solve by using the Quadratic Formula: \(5{n}^{2}+4n-4=0\) .

\(n=\frac{-2+2\sqrt{6}}{5},n=\frac{-2-2\sqrt{6}}{5}\)

Did you get it?

When we substitute a, b, and c into the Quadratic Formula and the radicand is negative, the quadratic equation will have imaginary or complex solutions. We will see this in the next example.

Example 4

Solve by using the Quadratic Formula: \(3{p}^{2}+2p+9=0.\)

Substitute a, b, and c into the formula; a negative radicand signals complex solutions.

Table 4
A quadratic equation displayed as 3p^2 + 2p + 9 = 0.
This equation is in standard formTwo quadratic equations are displayed: the general form ax^2 + bx + c = 0 in red and a specific example 3p^2 + 2p + 9 = 0 in black.
Identify the values of \(a,b,c.\)The image displays the values of three variables: a=3, b=2, and c=9, written in different colors on a white background.
Write the Quadratic Formula.The quadratic formula, an algebraic formula that provides the solution(s) for a quadratic equation, where p equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.
Then substitute in the values of \(a,b,c\) .A mathematical expression for 'p' using the quadratic formula, featuring numbers 2, 3, and 9. It shows p = [-2 ×1 sqrt(2^2 - 4 * 3 * 9)] / (2 * 3).
Simplify.A mathematical equation shows p equals a fraction where the numerator is -2 plus or minus the square root of (4 minus 108), and the denominator is 6.
A mathematical equation showing the variable p is equal to a fraction where the numerator is -2 plus or minus the square root of -104, and the denominator is 6.
Simplify the radical using complex numbers.The image displays a mathematical equation for the variable 'p', showing it as a complex number: p = (-2 A plus-minus sign sqrt(104) i) / 6.
Simplify the radical.The image shows the mathematical equation for p, which is equal to a fraction with -2 plus or minus 2 times the square root of 26i in the numerator, and 6 in the denominator.
Factor the common factor in the numerator.A mathematical equation displaying 'p' equals 2 multiplied by the quantity of negative 1 plus or minus the square root of 26i, all divided by 6.
Remove the common factors.A mathematical equation shows p equals the fraction with a numerator of -1 plus or minus the square root of 26 multiplied by i, and a denominator of 3.
Rewrite in standard \(a+bi\) form.The image shows the complex solution for p, expressed as p = -1/3 plus or minus (square root of 26 multiplied by i) all divided by 3, written in a clear, standard mathematical notation.
Write as two solutions.The image displays the two complex conjugate values for 'p': p = -1/3 + (sqrt(26)i)/3 and p = -1/3 - (sqrt(26)i)/3.
Try It #7

Solve by using the Quadratic Formula: \(4{a}^{2}-2a+8=0\) .

\(a=\frac{1}{4}+\frac{\sqrt{31}}{4}i,\,\,\,a=\frac{1}{4}-\frac{\sqrt{31}}{4}i\)

Did you get it?
Try It #8

Solve by using the Quadratic Formula: \(5{b}^{2}+2b+4=0\) .

\(b=-\frac{1}{5}+\frac{\sqrt{19}}{5}i,\,\,\,b=-\frac{1}{5}-\frac{\sqrt{19}}{5}i\)

Did you get it?

Remember, to use the Quadratic Formula, the equation must be written in standard form, ax2 + bx + c = 0. Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.

Example 5

Solve by using the Quadratic Formula: \(x(x+6)+4=0.\)

Distribute to put the equation into standard form before identifying a, b, and c.

Our first step is to get the equation in standard form.

Table 5
A mathematical equation is displayed, showing x multiplied by the quantity (x plus 6), plus 4, which equals 0. The equation is x(x+6) + 4 = 0.
Distribute to get the equation in standard form.A mathematical equation is displayed, showing a quadratic expression set equal to zero: x^2 + 6x + 4 = 0. The equation is rendered in a clear, digital font on a plain white background.
This equation is now in standard formTwo quadratic equations are displayed: 'ax^2 + bx + c = 0' in red text, followed by 'x^2 + 6x + 4 = 0' in black text below it.
Identify the values of \(a,b,c.\)The image displays the values of three variables: a=1, b=6, and c=4, with each variable and its value presented in a distinct color, suggesting a mathematical or programming context.
Write the Quadratic Formula.The quadratic formula: x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.
Then substitute in the values of \(a,b,c\) .Mathematical equation showing the quadratic formula with 'b=6' (red), 'a=1' (light blue), and 'c=4' (yellow) color-coded and substituted into the expression x = -(6) ×plusminus ×sqrt{(6)^2 - 4 * 1 * 4} / (2 * 1).
Simplify.The image shows a step in solving a quadratic equation, with x equal to a fraction where the numerator is -6 plus or minus the square root of (36 minus 16), and the denominator is 2.
A mathematical equation is displayed: x equals the fraction with a numerator of -6 plus or minus the square root of 20, and a denominator of 2.
Simplify the radical.A mathematical equation shows x equals the fraction with a numerator of -6 plus or minus 2 times the square root of 5, all divided by 2.
Factor the common factor in the numerator.A mathematical equation shows x equals 2 times the quantity of negative 3 plus or minus the square root of 5, all divided by 2. This represents a solution to a quadratic equation.
Remove the common factors.A mathematical equation is displayed, showing 'x = -3 ×1 ×5' in black font against a white background.
Write as two solutions.The image displays the solutions to a quadratic equation, showing x = -3 + sqrt(5) and x = -3 - sqrt(5).
Check:
We leave the check for you!
Try It #9

Solve by using the Quadratic Formula: \(x(x+2)-5=0.\)

\(x=-1+\sqrt{6},x=-1-\sqrt{6}\)

Did you get it?
Try It #10

Solve by using the Quadratic Formula: \(3y(y-2)-3=0.\)

\(y=1+\sqrt{2},y=1-\sqrt{2}\)

Did you get it?

When we solved linear equations, if an equation had too many fractions we cleared the fractions by multiplying both sides of the equation by the LCD. This gave us an equivalent equation—without fractions— to solve. We can use the same strategy with quadratic equations.

Example 6

Solve by using the Quadratic Formula: \(\frac{1}{2}{u}^{2}+\frac{2}{3}u=\frac{1}{3}.\)

Clear the fractions by multiplying both sides by the LCD before writing the equation in standard form.

Our first step is to clear the fractions.

Table 6
A quadratic equation is displayed on a white background, reading one-half u squared plus two-thirds u equals one-third.
Multiply both sides by the LCD, 6, to clear the fractions.A mathematical equation is displayed, showing 6 multiplied by the sum of one-half u squared and two-thirds u, equaling 6 multiplied by one-third.
Multiply.The image displays the mathematical quadratic equation 3u^2 + 4u = 2, presented in a clear, dark gray font against a plain white background.
Subtract 2 to get the equation in standard form.Two lines of equations are displayed. The first line is 'ax^2 + bx + c = 0' in red. The second line, in black, is '3u^2 + 4u - 2 = 0'.
Identify the values of a, b, and c.The image displays the equations a = 3 in light blue, b = 4 in red, and c = 2 in yellow-green, all arranged horizontally on a white background.
Write the Quadratic Formula.The image displays the quadratic formula, an algebraic formula used to solve quadratic equations of the form ax^2 + bx + c = 0, where 'u' represents the variable.
Then substitute in the values of a, b, and c.An example of the quadratic formula in use, with coefficients 'b=4', 'a=3', and 'c=-2' substituted to find the value of 'u'. The values are color-coded for clarity.
Simplify.A mathematical equation, 'u = (-4 plus or minus sqrt(16 + 24)) / 6'
A mathematical equation shows 'u equals negative four plus or minus the square root of forty, all divided by six'.
Simplify the radical.The image shows a mathematical equation where 'u' equals the fraction whose numerator is '-4 plus or minus 2 times the square root of 10' and whose denominator is '6'.
Factor the common factor in the numerator.A mathematical equation is displayed where 'u' is equal to 2 multiplied by the quantity of -2 plus or minus the square root of 10, all divided by 6, set against a plain white background.
Remove the common factors.A mathematical equation is displayed, showing u equals a fraction: negative 2 plus or minus the square root of 10, all divided by 3.
Rewrite to show two solutions.Two mathematical expressions are shown for the variable u. The first is u equals negative two plus the square root of ten, all divided by three. The second is u equals negative two minus the square root of ten, all divided by three.
Check:
We leave the check for you!
Try It #11

Solve by using the Quadratic Formula: \(\frac{1}{4}{c}^{2}-\frac{1}{3}c=\frac{1}{12}\) .

\(c=\frac{2+\sqrt{7}}{3},\,\,\,c=\frac{2-\sqrt{7}}{3}\)

Did you get it?
Try It #12

Solve by using the Quadratic Formula: \(\frac{1}{9}{d}^{2}-\frac{1}{2}d=-\frac{1}{3}\) .

\(d=\frac{9+\sqrt{33}}{4},\,\text{d}=\frac{9-\sqrt{33}}{4}\)

Did you get it?

Think about the equation (x − 3)2 = 0. We know from the Zero Product Property that this equation has only one solution,
x = 3.

We will see in the next example how using the Quadratic Formula to solve an equation whose standard form is a perfect square trinomial equal to 0 gives just one solution. Notice that once the radicand is simplified it becomes 0 , which leads to only one solution.

Example 7

Solve by using the Quadratic Formula: \(4{x}^{2}-20x=-25.\)

Move all terms to standard form; watch what happens to the radicand once you simplify it.

Table 7
A quadratic equation, 4x^2 - 20x = -25, is displayed on a white background.
Add 25 to get the equation in standard form.A quadratic equation in its general form (ax² + bx + c = 0) is shown, followed by a specific example (4x² - 20x + 25 = 0).
Identify the values of a, b, and c.The image displays mathematical variable assignments: a = 4, b = 20, c = 25, shown in blue, red, and yellow text, respectively, against a white background.
Write the quadratic formula.The quadratic formula, x = (-b ×1 sqrt(b^2 - 4ac)) / 2a, is displayed in black text on a white background.
Then substitute in the values of a, b, and c.An image showing the quadratic formula x = -b ± sqrt(b^2 - 4ac) / 2a with values b=-20, a=4, and c=25 substituted, displaying an intermediate step in solving a quadratic equation.
Simplify.A mathematical equation shows x equals the fraction with a numerator of 20 plus or minus the square root of 400 minus 400, and a denominator of 8.
A mathematical equation showing x equals twenty plus or minus the square root of zero, all divided by eight.
Simplify the radical.A mathematical equation shows 'x = 20/8' written in black against a white background.
Simplify the fraction.The image displays the mathematical equation 'x = 5/2' in black text against a plain white background, showing the value of the variable x as a fraction.
Check:
We leave the check for you!

Did you recognize that 4x2 − 20x + 25 is a perfect square trinomial. It is equivalent to (2x − 5)2? If you solve
4x2 − 20x + 25 = 0 by factoring and then using the Square Root Property, do you get the same result?

Try It #13

Solve by using the Quadratic Formula: \({r}^{2}+10r+25=0.\)

\(r=-5\)

Did you get it?
Try It #14

Solve by using the Quadratic Formula: \(25{t}^{2}-40t=-16.\)

\(t=\frac{4}{5}\)

Did you get it?

Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?

Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the discriminant.

DiscriminantIn the Quadratic Formula, x equals the quotient of negative b plus or minus the square root of b squared minus 4 times a times c and 2 a, the value under the radical, b squared minus 4 times a times c, is called the discriminant.

Let’s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.

Table 8
Quadratic Equation
(in standard form)		Discriminant
\({b}^{2}-4ac\)		Value of the DiscriminantNumber and Type of solutions
\(2{x}^{2}+9x-5=0\)\(\begin{array}{l}{9}^{2}-4·2(-5) \\ 121\end{array}\)+2 real
\(4{x}^{2}-20x+25=0\)\(\begin{array}{l}{(-20)}^{2}-4·4·25 \\ 0\end{array}\)01 real
\(3{p}^{2}+2p+9=0\)\(\begin{array}{l}{2}^{2}-4·3·9 \\ \\ -104\end{array}\)−2 complex
When the value under the radical in the Quadratic Formula, the discriminant, is positive, the equation has two real solutions. When the value under the radical in the Quadratic Formula, the discriminant, is zero, the equation has one real solution. When the value under the radical in the Quadratic Formula, the discriminant, is negative, the equation has two complex solutions.
Using the Discriminant, b2 − 4ac, to Determine the Number and Type of Solutions of a Quadratic Equation

For a quadratic equation of the form ax2 + bx + c = 0, \(a\ne 0,\)

  • If b2 − 4ac > 0, the equation has 2 real solutions.
  • if b2 − 4ac = 0, the equation has 1 real solution.
  • if b2 − 4ac < 0, the equation has 2 complex solutions.
Example 8

Determine the number of solutions to each quadratic equation.

ⓐ \(3{x}^{2}+7x-9=0\) ⓑ \(5{n}^{2}+n+4=0\) ⓒ \(9{y}^{2}-6y+1=0.\)

For each equation, identify a, b, c and compute the discriminant b² − 4ac, then check its sign.

To determine the number of solutions of each quadratic equation, we will look at its discriminant.

ⓐ

Table 9
\(3{x}^{2}+7x-9=0\)
The equation is in standard form, identify a, b, and c.\(a=3,\,b=7,\,c=-9\)
Write the discriminant.\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.\({(7)}^{2}-4·3·(-9)\)
Simplify.\(49+108\)
\(157\)

Since the discriminant is positive, there are 2 real solutions to the equation.

ⓑ

Table 10
\(5{n}^{2}+n+4=0\)
The equation is in standard form, identify a, b, and c.\(a=5,\,b=1,\,c=4\)
Write the discriminant.\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.\({(1)}^{2}-4·5·4\)
Simplify.\(1-80\)
\(-79\)

Since the discriminant is negative, there are 2 complex solutions to the equation.

ⓒ

Table 11
\(9{y}^{2}-6y+1=0\)
The equation is in standard form, identify a, b, and c.\(a=9,b=-6,c=1\)
Write the discriminant.\({b}^{2}-4ac\)
Substitute in the values of a, b, and c.\({(-6)}^{2}-4·9·1\)
Simplify.\(36-36\)
\(0\)

Since the discriminant is 0, there is 1 real solution to the equation.

Try It #15

Determine the numberand type of solutions to each quadratic equation.

ⓐ \(8{m}^{2}-3m+6=0\) ⓑ \(5{z}^{2}+6z-2=0\) ⓒ \(9{w}^{2}+24w+16=0.\)

ⓐ 2 complex solutions; ⓑ 2 real solutions; ⓒ 1 real solution

Did you get it?
Try It #16

Determine the number and type of solutions to each quadratic equation.

ⓐ \({b}^{2}+7b-13=0\) ⓑ \(5{a}^{2}-6a+10=0\) ⓒ \(4{r}^{2}-20r+25=0.\)

ⓐ 2 real solutions; ⓑ 2 complex solutions; ⓒ 1 real solution

Did you get it?

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We summarize the four methods that we have used to solve quadratic equations below.

Methods for Solving Quadratic Equations
  • Factoring
  • Square Root Property
  • Completing the Square
  • Quadratic Formula

Given that we have four methods to use to solve a quadratic equation, how do you decide which one to use? Factoring is often the quickest method and so we try it first. If the equation is \(a{x}^{2}=k\) or \(a{(x-h)}^{2}=k\) we use the Square Root Property. For any other equation, it is probably best to use the Quadratic Formula. Remember, you can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method.

What about the method of Completing the Square? Most people find that method cumbersome and prefer not to use it. We needed to include it in the list of methods because we completed the square in general to derive the Quadratic Formula. You will also use the process of Completing the Square in other areas of algebra.

Identify the most appropriate method to solve a quadratic equation.
  • Try Factoring first. If the quadratic factors easily, this method is very quick.
  • Try the Square Root Property next. If the equation fits the form \(a{x}^{2}=k\) or \(a{(x-h)}^{2}=k,\) it can easily be solved by using the Square Root Property.
  • Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

The next example uses this strategy to decide how to solve each quadratic equation.

Example 9

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ \(5{z}^{2}=17\) ⓑ \(4{x}^{2}-12x+9=0\) ⓒ \(8{u}^{2}+6u=11.\)

Check whether each equation fits the form ax² = k, is a perfect square trinomial, or fits neither pattern.

ⓐ
\(\begin{array}{l} \\ \\ \\ \,5{z}^{2}=17\end{array}\)

Since the equation is in the \(a{x}^{2}=k,\) the most appropriate method is to use the Square Root Property.

ⓑ
\(\begin{array}{l} \\ \\ \\ \,4{x}^{2}-12x+9=0\end{array}\)

We recognize that the left side of the equation is a perfect square trinomial, and so factoring will be the most appropriate method.

ⓒ

Table 12
\(8{u}^{2}+6u=11\)
Put the equation in standard form.\(8{u}^{2}+6u-11=0\)

While our first thought may be to try factoring, thinking about all the possibilities for trial and error method leads us to choose the Quadratic Formula as the most appropriate method.

Try It #17

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ \({x}^{2}+6x+8=0\) ⓑ \({(n-3)}^{2}=16\) ⓒ \(5{p}^{2}-6p=9.\)

ⓐ factoring; ⓑ Square Root Property; ⓒ Quadratic Formula

Did you get it?
Try It #18

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ \(8{a}^{2}+3a-9=0\) ⓑ \(4{b}^{2}+4b+1=0\) ⓒ \(5{c}^{2}=125.\)

ⓐ Quadratic Forumula;
ⓑ Factoring or Square Root Property ⓒ Square Root Property

Did you get it?
Media

Access these online resources for additional instruction and practice with using the Quadratic Formula.

Key Concepts

Section Exercises

Practice Makes Perfect

Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

4

\(4{m}^{2}+m-3=0\)

\(m=-1,m=\frac{3}{4}\)

5

\(4{n}^{2}-9n+5=0\)

6

\(2{p}^{2}-7p+3=0\)

\(p=\frac{1}{2},p=3\)

7

\(3{q}^{2}+8q-3=0\)

8

\({p}^{2}+7p+12=0\)

\(p=-4,p=-3\)

9

\({q}^{2}+3q-18=0\)

10

\({r}^{2}-8r=33\)

\(r=-3,r=11\)

11

\({t}^{2}+13t=-40\)

12

\(3{u}^{2}+7u-2=0\)

\(u=\frac{-7\pm \sqrt{73}}{6}\)

13

\(2{p}^{2}+8p+5=0\)

14

\(2{a}^{2}-6a+3=0\)

\(a=\frac{3\pm \sqrt{3}}{2}\)

15

\(5{b}^{2}+2b-4=0\)

16

\({x}^{2}+8x-4=0\)

\(x=-4\pm 2\sqrt{5}\)

17

\({y}^{2}+4y-4=0\)

18

\(3{y}^{2}+5y-2=0\)

\(y=-2,y=\frac{1}{3}\)

19

\(6{x}^{2}+2x-20=0\)

20

\(2{x}^{2}+3x+3=0\)

\(x=-\frac{3}{4}\pm \frac{\sqrt{15}}{4}i\)

21

\(2{x}^{2}-x+1=0\)

22

\(8{x}^{2}-6x+2=0\)

\(x=\frac{3}{8}\pm \frac{\sqrt{7}}{8}i\)

23

\(8{x}^{2}-4x+1=0\)

24

\((v+1)(v-5)-4=0\)

\(v=2\pm \sqrt{13}\)

25

\((x+1)(x-3)=2\)

26

\((y+4)(y-7)=18\)

\(y=\frac{3\pm \sqrt{193}}{2}\)

27

\((x+2)(x+6)=21\)

28

\(\frac{1}{3}{m}^{2}+\frac{1}{12}m=\frac{1}{4}\)

\(m=-1,m=\frac{3}{4}\)

29

\(\frac{1}{3}{n}^{2}+n=-\frac{1}{2}\)

30

\(\frac{3}{4}{b}^{2}+\frac{1}{2}b=\frac{3}{8}\)

\(b=\frac{-2\pm \sqrt{22}}{6}\)

31

\(\frac{1}{9}{c}^{2}+\frac{2}{3}c=3\)

32

\(16{c}^{2}+24c+9=0\)

\(c=-\frac{3}{4}\)

33

\(25{d}^{2}-60d+36=0\)

34

\(25{q}^{2}+30q+9=0\)

\(q=-\frac{3}{5}\)

35

\(16{y}^{2}+8y+1=0\)

Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation

In the following exercises, determine the number of real solutions for each quadratic equation.

36

ⓐ \(4{x}^{2}-5x+16=0\) ⓑ \(36{y}^{2}+36y+9=0\) ⓒ \(6{m}^{2}+3m-5=0\)

ⓐ \(\text{no real solutions}\) ⓑ \(1\)
ⓒ \(2\)

37

ⓐ \(9{v}^{2}-15v+25=0\) ⓑ \(100{w}^{2}+60w+9=0\) ⓒ \(5{c}^{2}+7c-10=0\)

38

ⓐ \({r}^{2}+12r+36=0\) ⓑ \(8{t}^{2}-11t+5=0\) ⓒ \(3{v}^{2}-5v-1=0\)

ⓐ \(1\) ⓑ \(\text{no real solutions}\)
ⓒ \(2\)

39

ⓐ \(25{p}^{2}+10p+1=0\) ⓑ \(7{q}^{2}-3q-6=0\) ⓒ \(7{y}^{2}+2y+8=0\)

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

40


ⓐ \({x}^{2}-5x-24=0\)
ⓑ \({(y+5)}^{2}=12\)
ⓒ \(14{m}^{2}+3m=11\)


ⓐ \(\text{factor}\)
ⓑ \(\text{square root}\)
ⓒ \(\text{Quadratic Formula}\)

41


ⓐ \({(8v+3)}^{2}=81\)
ⓑ \({w}^{2}-9w-22=0\)
ⓒ \(4{n}^{2}-10n=6\)

42


ⓐ \(6{a}^{2}+14a=20\)
ⓑ \({(x-\frac{1}{4})}^{2}=\frac{5}{16}\)
ⓒ \({y}^{2}-2y=8\)


ⓐ \(\text{Quadratic Formula}\)
ⓑ \(\text{square root}\)
ⓒ \(\text{factor}\)

43


ⓐ \(8{b}^{2}+15b=4\)
ⓑ \(\frac{5}{9}{v}^{2}-\frac{2}{3}v=1\)
ⓒ \({(w+\frac{4}{3})}^{2}=\frac{2}{9}\)

Writing Exercises

44

Solve the equation \({x}^{2}+10x=120\)

ⓐ by completing the square

ⓑ using the Quadratic Formula

ⓒ Which method do you prefer? Why?

Answers will vary.

45

Solve the equation \(12{y}^{2}+23y=24\)

ⓐ by completing the square

ⓑ using the Quadratic Formula

ⓒ Which method do you prefer? Why?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can solve quadratic equations using the quadratic formula.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can use the discriminant to predict the number of solutions of a quadratic equation.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can identify the most appropriate method to use to solve a quadratic equation.” “Confidently,” “with some help,” or “No, I don’t get it.”

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Glossary

discriminant
In the Quadratic Formula, \(x=\frac{\text{-}b\pm \sqrt{{b}^{2}-4ac}}{2a},\) the quantity b2 − 4ac is called the discriminant.