MX Algebra Solve Quadratic Equations by Completing the Square

Section 9.2Solve Quadratic Equations by Completing the Square

Definition

Before you get started, take this readiness quiz.

1

Expand: \({(x+9)}^{2}.\)
If you missed this problem, review Example 8.

\({x}^{2}+18x+81\)

Definition
2

Factor \({y}^{2}-14y+49.\)
If you missed this problem, review Example 1.

\((y-7{)}^{2}\)

Definition
3

Factor \(5{n}^{2}+40n+80.\)
If you missed this problem, review Example 6.

\(5(n+4{)}^{2}\)

So far we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called completing the square, which is important for our work on conics later.

Complete the Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation (y − 7)2 = 12 because the left side was a perfect square.

\[\begin{array}{lll}{(y-7)}^{2} & = & 12 \\ y-7 & = & \pm \sqrt{12} \\ y-7 & = & \pm 2\sqrt{3} \\ y & = & 7\pm 2\sqrt{3}\end{array}\]

We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form \({(x-k)}^{2}\) in order to use the Square Root Property.

\[\begin{array}{lll}{x}^{2}-10x+25 & = & 18 \\ {(x-5)}^{2} & = & 18\end{array}\]

What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?

Let’s look at two examples to help us recognize the patterns.

\[\begin{array}{llll}{(x+9)}^{2} & & & \,{(y-7)}^{2} \\ (x+9)(x+9) & & & \,(y-7)(y-7) \\ {x}^{2}+9x+9x+81 & & & \,{y}^{2}-7y-7y+49 \\ {x}^{2}+18x+81 & & & \,{y}^{2}-14y+49\end{array}\]

We restate the patterns here for reference.

Binomial Squares Pattern

If a and b are real numbers,

Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.

We can use this pattern to “make” a perfect square.

We will start with the expression x2 + 6x. Since there is a plus sign between the two terms, we will use the (a + b)2 pattern, a2 + 2ab + b2 = (a + b)2.

The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6x plus an unknown to allow a comparison of the corresponding terms of the expressions.

We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find b. But first we start with determining a. Notice that the first term of x2 + 6x is a square, x2. This tells us that a = x.

The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 x b + b squared. Note that x has been substituted for a in the second equation and compare corresponding terms.

What number, b, when multiplied with 2x gives 6x? It would have to be 3, which is \(\frac{1}{2}(6).\) So b = 3.

The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 times 3 times x plus an unknown value to help compare terms.

Now to complete the perfect square trinomial, we will find the last term by squaring b, which is 32 = 9.

The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6 x plus 9.

We can now factor.

The factored expression, the square of a plus b, is shown over the square of the expression x + 3.

So we found that adding 9 to x2 + 6x ‘completes the square’, and we write it as (x + 3)2.

Complete a square of \({x}^{2}+bx.\)
  • Identify b, the coefficient of x.
  • Find \({(\frac{1}{2}b)}^{2},\) the number to complete the square.
  • Add the \({(\frac{1}{2}b)}^{2}\) to x2 + bx.
  • Factor the perfect square trinomial, writing it as a binomial squared.
Example 1

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

ⓐ \({x}^{2}-26x\) ⓑ \({y}^{2}-9y\) ⓒ \({n}^{2}+\frac{1}{2}n\)

For each expression, take half the coefficient of the linear term and square it to find the missing constant.

ⓐ

Table 1
Two algebraic expressions are displayed: 'x² - bx' in red text, followed by 'x² - 26x' in black text directly below it.
The coefficient of \(x\) is −26.
\(\begin{array}{l} \\ \\ \\ \text{Find}\,{(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}·(-26))}^{2} \\ {(13)}^{2} \\ 169\end{array}\)
Add 169 to the binomial to complete the square.The image shows the algebraic expression x^2 - 26x + 169, which is a quadratic trinomial. This expression is a perfect square trinomial, factoring to (x - 13)^2.
Factor the perfect square trinomial, writing it as
a binomial squared.		A mathematical expression showing the quantity (x-13) raised to the power of 2, enclosed in parentheses.

ⓑ

Table 2
Two algebraic expressions are shown: x^2 - bx in red, and y^2 - 9y in black, stacked vertically.
The coefficient of \(y\) is \(-9\) .
\(\begin{array}{l} \\ \\ \\ \text{Find}\,{(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}·(-9))}^{2} \\ {(\text{-}\frac{9}{2})}^{2} \\ \frac{81}{4}\end{array}\)
Add \(\frac{81}{4}\) to the binomial to complete the square.A mathematical expression shown in black text on a white background, which reads y squared minus 9y plus 81 over 4.
Factor the perfect square trinomial, writing it as
a binomial squared.		(y - 9/2)^2

ⓒ

Table 3
Two lines of algebraic expressions are shown on a white background. The top line reads 'x^2 + bx' in red text. The bottom line reads 'n^2 + (1/2)n' in black text.
The coefficient of \(n\) is \(\frac{1}{2}.\)
\(\begin{array}{l} \\ \\ \\ \text{Find}\,{(\frac{1}{2}b)}^{2}. \\ {(\frac{1}{2}·\frac{1}{2})}^{2} \\ {(\frac{1}{4})}^{2} \\ \frac{1}{16}\end{array}\)
Add \(\frac{1}{16}\) to the binomial to complete the square.A mathematical expression displays n squared plus one-half n plus one-sixteenth, set against a plain white background.
Rewrite as a binomial square.A mathematical expression showing the quantity 'n plus one-fourth' enclosed in parentheses, all raised to the power of two: (n + 1/4)^2.
Try It #1

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

ⓐ \({a}^{2}-20a\) ⓑ \({m}^{2}-5m\) ⓒ \({p}^{2}+\frac{1}{4}p\)

ⓐ \({(a-10)}^{2}\) ⓑ \({(b-\frac{5}{2})}^{2}\)
ⓒ \({(p+\frac{1}{8})}^{2}\)

Did you get it?
Try It #2

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

ⓐ \({b}^{2}-4b\) ⓑ \({n}^{2}+13n\) ⓒ \({q}^{2}-\frac{2}{3}q\)

ⓐ \({(b-2)}^{2}\) ⓑ \({(n+\frac{13}{2})}^{2}\)
ⓒ \({(q-\frac{1}{3})}^{2}\)

Did you get it?

Solve Quadratic Equations of the Form x2 + bx + c = 0 by Completing the Square

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.

For example, if we start with the equation x2 + 6x = 40, and we want to complete the square on the left, we will add 9 to both sides of the equation.

Table 4
The image displays the quadratic equation x squared plus 6x equals 40.
An incomplete quadratic equation is shown, x squared plus 6x plus an empty blank space on the left side, which equals 40 plus another empty blank space on the right side.
A mathematical equation, x^2 + 6x + 9 = 40 + 9, with the number 9 highlighted in red, indicating it has been added to both sides to complete the square.
Add 9 to both sides to complete the square.A mathematical equation is displayed: (x + 3)' = 49. The equation involves an algebraic expression squared equal to a numerical value, representing a quadratic equation in a specific format.

Now the equation is in the form to solve using the Square Root Property! Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.

Example 2How to Solve a Quadratic Equation of the Form \({x}^{2}+bx+c=0\) by Completing the Square

Solve by completing the square: \({x}^{2}+8x=48.\)

Take half of the coefficient of x, square it, and add that value to both sides.

Step 1 is to isolate the variable terms on one side and the constant terms on the other. This equation, x squared plus 8 x equals 48 already has all variable terms on the left. Note that the leading coefficient is 1, so b equals 8. In step 2, find the expression one half times b, squared, the number needed to complete the square. Add this value to both sides of the equation. Take half of 8 and square it. The square of one half times 8 equals 16, so add 16 to BOTH sides of the equation. The equation becomes x squared plus 8 x plus 16 equals 48 plus 16. In step 3, factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right. Factor x squared plus 8 x plus 16 on the left side. Add 48+16 on the right side. The equation becomes the square of x plus 4 equals 64. Step 4 is to use the Square Root Property. Take the square root of both sides of the equation to yield x plus 4 equals the positive or negative square root of 64. In step 5, simplify the radical and then solve the two resulting equations. X plus 4 equals positive 8 or negative 8. If x plus 4 equals 8, then x equals 4. If x plus 4 equals negative 8, then x equals negative 12. Finally, step 6, check the solutions. Put each answer in the original equation to check. First substitute x equals 4. We need to show that 4 squared plus 8 times 4 equals 48. Simplify. The expression 4 squared plus 8 times 4 is equivalent to 16 plus 32, or 48. X equals 4 is a solution. Next substitute x equals negative 12 into the original equation, x squared plus 8 x equals 48. The square of negative 12 plus 8 times negative 12 equals 144 minus 96, or 48. X equals negative 12 is also a solution.
Try It #3

Solve by completing the square: \({x}^{2}+4x=5.\)

\(x=-5,x=1\)

Did you get it?
Try It #4

Solve by completing the square: \({y}^{2}-10y=-9.\)

\(y=1,y=9\)

Did you get it?

The steps to solve a quadratic equation by completing the square are listed here.

Solve a quadratic equation of the form \({x}^{2}+bx+c=0\) by completing the square.
  • Isolate the variable terms on one side and the constant terms on the other.
  • Find \({(\frac{1}{2}\,·\,b)}^{2},\) the number needed to complete the square. Add it to both sides of the equation.
  • Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right
  • Use the Square Root Property.
  • Simplify the radical and then solve the two resulting equations.
  • Check the solutions.

When we solve an equation by completing the square, the answers will not always be integers.

Example 3

Solve by completing the square: \({x}^{2}+4x=-21.\)

Complete the square as before; watch for a negative value once you reach the Square Root Property step.

Table 5
Two math expressions are shown: the general quadratic form x² + bx + c in red, and a specific quadratic equation x² + 4x = -21 in black on a white background.
The variable terms are on the left side.
Take half of 4 and square it.		An algebraic equation showing the initial step of completing the square: x^2 + 4x + [blank line with (1/2 * 4)^2 in red below it] = -21.
\({(\frac{1}{2}(4))}^{2}=4\)
Add 4 to both sides.A mathematical equation showing the step of adding 4 to both sides: x^2 + 4x + 4 = -21 + 4, as part of completing the square to solve for x.
Factor the perfect square trinomial,
writing it as a binomial squared.		The image displays the quadratic equation (x + 2)^2 = -17, which has no real solutions.
Use the Square Root Property.A mathematical equation is displayed on a white background: x + 2 = ±sqrt(-17). This equation involves a variable 'x', constants, an equality sign, and the plus-minus square root of a negative number, indicating complex solutions.
Simplify using complex numbers.Equation showing x plus 2 equals positive or negative square root of 17 times i. This represents complex solutions for x.
Subtract 2 from each side.The image displays a mathematical equation for x: x = -2 ± sqrt(17)i, showing a solution with both real and imaginary components.
Rewrite to show two solutions.Two complex conjugate roots are displayed: x = -2 + sqrt(17)i and x = -2 - sqrt(17)i.
We leave the check to you.
Try It #5

Solve by completing the square: \({y}^{2}-10y=-35.\)

\(y=5\pm \sqrt{10}i\)

Did you get it?
Try It #6

Solve by completing the square: \({z}^{2}+8z=-19.\)

\(z=-4+\sqrt{3}i,\,\text{z}=-4-\sqrt{3}i\)

Did you get it?

In the previous example, our solutions were complex numbers. In the next example, the solutions will be irrational numbers.

Example 4

Solve by completing the square: \({y}^{2}-18y=-6.\)

Take half the coefficient of y, square it, and add it to both sides; the result won't simplify to a whole number.

Table 6
Two algebraic expressions are shown: one on the top reads 'x^2 - bx c' with 'c' appearing slightly separate, and the other below it reads 'y^2 - 18y = -6'.
The variable terms are on the left side.
Take half of \(-18\) and square it.
\({(\frac{1}{2}(-18))}^{2}=81\)A mathematical equation y^2 - 18y + blank = -6 is shown, with a red hint below the blank indicating (1/2 * -18)^2, illustrating the process of completing the square to solve a quadratic equation.
Add 81 to both sides.An equation: y^2 - 18y + 81 = -6 + 81. The value 81 is highlighted in red, indicating its addition to both sides, a common step in solving quadratic equations by completing the square.
Factor the perfect square trinomial,
writing it as a binomial squared.		A mathematical equation is displayed, showing (y-9)^2 = 75. The equation involves a variable y, subtraction, exponentiation, and equality to a constant.
Use the Square Root Property.A mathematical equation is displayed, showing y - 9 = +/- sqrt(75).
Simplify the radical.A mathematical equation shows 'y minus 9 equals plus or minus 5 times the square root of 3'.
Solve for \(y\) .A mathematical expression showing y equals 9 plus or minus 5 times the square root of 3.
Check.
Verifying solutions for the quadratic equation y² - 18y = -6 by substituting y with (9 + 5√3) and (9 - 5√3), both yielding -6 = -6.

Another way to check this would be to use a calculator. Evaluate \({y}^{2}-18y\) for both of the solutions. The answer should be \(-6.\)

Try It #7

Solve by completing the square: \({x}^{2}-16x=-16.\)

\(x=8+4\sqrt{3},\,\,\,x=8-4\sqrt{3}\)

Did you get it?
Try It #8

Solve by completing the square: \({y}^{2}+8y=11.\)

\(y=-4+3\sqrt{3},\,\text{y}=-4-3\sqrt{3}\)

Did you get it?

We will start the next example by isolating the variable terms on the left side of the equation.

Example 5

Solve by completing the square: \({x}^{2}+10x+4=15.\)

First move the constant term to the right side, then complete the square as usual.

Table 7
A mathematical equation is displayed on a white background. The equation reads: x squared plus 10x plus 4 equals 15.
Isolate the variable terms on the left side.
Subtract 4 to get the constant terms on the right side.		A quadratic equation displayed against a plain white background, reading 'x^2 + 10x = 11'.
Take half of 10 and square it.
\({(\frac{1}{2}(10))}^{2}=25\)A mathematical equation illustrating a step in completing the square: x^2 - 10x + blank = 11, where the blank is represented by a fraction bar with an empty numerator and a denominator of (1/2 * 10)^2 in red.
Add 25 to both sides.A quadratic equation shown as x squared plus 10x plus 25 equals 11 plus 25, with the number 25 highlighted in red on both sides of the equation.
Factor the perfect square trinomial, writing it as
a binomial squared.		A mathematical equation shows '(x + 5)^2 = 36' centered on a white background.
Use the Square Root Property.A mathematical equation is displayed, reading 'x + 5 = +/- square root 36', showing a step in solving for x in an algebraic problem involving a square root.
Simplify the radical.A mathematical equation is displayed, showing 'x + 5 = 6' with a plus-minus symbol before the 6.
Solve for x.A mathematical equation is displayed on a white background: x = -5 ×1 6. This represents two possible values for x: x = -5 + 6 and x = -5 - 6.
Rewrite to show two solutions.The image displays two algebraic expressions for the variable x: x = -5 + 6 and x = -5 - 6, indicating two different calculations.
Solve the equations.Two mathematical equations are displayed: x=1 and x=-11, set against a plain white background.
Check:

Verification of the quadratic equation x^2 + 10x + 4 = 15 by substituting x=1 and x=-11, showing that both values yield 15=15 and are therefore correct solutions.
Try It #9

Solve by completing the square: \({a}^{2}+4a+9=30.\)

\(a=-7,a=3\)

Did you get it?
Try It #10

Solve by completing the square: \({b}^{2}+8b-4=16.\)

\(b=-10,b=2\)

Did you get it?

To solve the next equation, we must first collect all the variable terms on the left side of the equation. Then we proceed as we did in the previous examples.

Example 6

Solve by completing the square: \({n}^{2}=3n+11.\)

Move all the variable terms to one side of the equation first, then complete the square.

Table 8
A mathematical equation is displayed on a white background: n squared equals three n plus eleven (n^2 = 3n + 11).
Subtract \(3n\) to get the variable terms on the left side.A mathematical equation is displayed on a white background, reading 'n squared minus 3n equals 11'.
Take half of \(-3\) and square it.
\({(\frac{1}{2}(-3))}^{2}=\frac{9}{4}\)A mathematical equation illustrating the process of completing the square: n^2 - 3n + (empty numerator) / ((1/2)*(-3))^2 = 11. The term to be added to complete the square is represented by the fraction.
Add \(\frac{9}{4}\) to both sides.An algebraic equation is displayed: n squared minus 3n plus 9 over 4 equals 11 plus 9 over 4. This likely represents a step in completing the square for a quadratic equation.
Factor the perfect square trinomial, writing it as
a binomial squared.		A mathematical equation shows (n - 3/2)^2 = 44/4 + 9/4. The expression on the left is a binomial squared with variable 'n' and constant 3/2. The right side is a sum of two fractions, both with a denominator of 4.
Add the fractions on the right side.A mathematical equation shown on a white background, which reads '(n - 3/2)^2 = 53/4'
Use the Square Root Property.A mathematical equation is displayed, showing 'n - 3/2 = ±√(53/4)' in black text on a white background.
Simplify the radical.A mathematical equation shows 'n minus 3 over 2 equals plus or minus the square root of 53 over 2'.
Solve for n.A mathematical equation displays 'n = 3/2 plus or minus the square root of 53, all divided by 2' in black text on a white background.
Rewrite to show two solutions.Two mathematical expressions for 'n' are shown: n = 3/2 + sqrt(53)/2, and n = 3/2 - sqrt(53)/2.
Check:
We leave the check for you!
Try It #11

Solve by completing the square: \({p}^{2}=5p+9.\)

\(p=\frac{5}{2}+\frac{\sqrt{61}}{2},\,\,\,p=\frac{5}{2}-\frac{\sqrt{61}}{2}\)

Did you get it?
Try It #12

Solve by completing the square: \({q}^{2}=7q-3.\)

\(q=\frac{7}{2}+\frac{\sqrt{37}}{2},\,\text{q}=\frac{7}{2}-\frac{\sqrt{37}}{2}\)

Did you get it?

Notice that the left side of the next equation is in factored form. But the right side is not zero. So, we cannot use the Zero Product Property since it says “If \(a\,·\,b=0,\) then a = 0 or b = 0.” Instead, we multiply the factors and then put the equation into standard form to solve by completing the square.

Example 7

Solve by completing the square: \((x-3)(x+5)=9.\)

Multiply out the binomials on the left first so the equation is in standard trinomial form.

Table 9
A mathematical equation is displayed against a white background: (x - 3)(x + 5) = 9. The equation involves a variable 'x' in two binomials that are multiplied together and set equal to 9.
We multiply the binomials on the left.A quadratic equation is displayed on a white background, reading 'x^2 + 2x - 15 = 9'.
Add 15 to isolate the constant terms on the right.A mathematical equation is displayed on a white background, reading 'x^2 + 2x = 24'. The equation features a quadratic term, a linear term, and a constant.
Take half of 2 and square it.
\({(\frac{1}{2}·(2))}^{2}=1\)An algebraic equation showing a step to complete the square. The expression is x^2 + 2x + a fractional term with a blank numerator and (1/2 * 2)^2 in the denominator, equaling 24.
Add 1 to both sides.The equation x^2 + 2x + 1 = 24 + 1, with the added '1' on each side emphasized in red.
Factor the perfect square trinomial, writing it as
a binomial squared.		A mathematical equation is displayed on a white background, reading '(x+1)^2 = 25' in black text.
Use the Square Root Property.A mathematical equation on a white background reads 'x + 1 = ±√25'.
Solve for x.A close-up shot of a math equation on a white background, which reads 'x = -1 ± 5'.
Rewrite to show two solutions.Two mathematical equations are displayed on a white background: 'x = -1 + 5' and 'x = -1 - 5'. These represent two possible solutions for the variable 'x'.
Simplify.Two mathematical expressions are displayed on a white background: 'x = 4,' and 'x = -6'.
Check:
We leave the check for you!
Try It #13

Solve by completing the square: \((c-2)(c+8)=11.\)

\(c=-9,c=3\)

Did you get it?
Try It #14

Solve by completing the square: \((d-7)(d+3)=56.\)

\(d=11,d=-7\)

Did you get it?

Solve Quadratic Equations of the Form ax2 + bx + c = 0 by Completing the Square

The process of completing the square works best when the coefficient of x2 is 1, so the left side of the equation is of the form x2 + bx + c. If the x2 term has a coefficient other than 1, we take some preliminary steps to make the coefficient equal to 1.

Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.

Example 8

Solve by completing the square: \(3{x}^{2}-12x-15=0.\)

Factor out the common coefficient of x² first so the leading coefficient becomes 1.

To complete the square, we need the coefficient of \({x}^{2}\) to be one. If we factor out the coefficient of \({x}^{2}\) as a common factor, we can continue with solving the equation by completing the square.

Table 10
A mathematical equation reads 3x^2 - 12x - 15 = 0.
Factor out the greatest common factor.A mathematical equation is displayed with the expression 3(x^2 - 4x - 5) = 0, indicating a quadratic equation to be solved.
Divide both sides by 3 to isolate the trinomial
with coefficient 1.		A mathematical equation shown as 3(x^2 - 4x - 5) / 3 = 0 / 3, demonstrating division by 3 on both sides of the equation.
Simplify.A quadratic equation, x^2 - 4x - 5 = 0, is displayed in a grayscale image. The equation is centered against a white background, representing a mathematical problem to be solved.
Add 5 to get the constant terms on the right side.A mathematical equation is displayed, showing 'x squared minus 4x equals 5' in a white background.
Take half of 4 and square it.
\({(\frac{1}{2}(-4))}^{2}=4\)An algebraic equation showing x^2 - 4x plus a blank numerator over a red (1/2 * 4)^2 term, set equal to 5. This illustrates a step in completing the square for a quadratic expression.
Add 4 to both sides.A step in solving x^2 - 4x = 5 by adding 4 to both sides to complete the square, resulting in x^2 - 4x + 4 = 5 + 4.
Factor the perfect square trinomial, writing it
as a binomial squared.		The image displays the algebraic equation (x-2)^2 = 9, centered on a white background.
Use the Square Root Property.A mathematical equation is displayed on a white background: x - 2 = plus or minus the square root of 9. This equation involves a variable, arithmetic operations, and a radical expression.
Solve for x.A mathematical equation 'x - 2 equals plus or minus 3'
Rewrite to show two solutions.The image displays two simple mathematical equations: x = 2 + 3 and x = 2 - 3, separated by a comma. These equations show two possible values for x, one from addition and one from subtraction.
Simplify.Two mathematical equations are displayed against a white background: 'x = 5,' and 'x = -1'.
Check:

Checking the solutions x=5 and x=-1 for the quadratic equation 3x² - 12x - 15 = 0. Both substitutions correctly result in 0=0, confirming they are valid roots.
Try It #15

Solve by completing the square: \(2{m}^{2}+16m+14=0.\)

\(m=-7,m=-1\)

Did you get it?
Try It #16

Solve by completing the square: \(4{n}^{2}-24n-56=8.\)

\(n=-2,n=8\)

Did you get it?

To complete the square, the coefficient of the x2 must be 1. When the leading coefficient is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient! This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.

Example 9

Solve by completing the square: \(2{x}^{2}-3x=20.\)

Divide every term by the coefficient of x² before completing the square.

To complete the square we need the coefficient of \({x}^{2}\) to be one. We will divide both sides of the equation by the coefficient of x2. Then we can continue with solving the equation by completing the square.

Table 11
A mathematical equation is displayed, reading '2x^2 - 3x = 20' in black text against a white background.
Divide both sides by 2 to get the
coefficient of \({x}^{2}\) to be 1.		A mathematical equation is shown with the expression (2x^2 - 3x) / 2 = 20 / 2, where both sides of the equation are divided by 2.
Simplify.A mathematical equation is displayed, showing 'x squared minus three halves x equals ten' in black text on a white background.
Take half of \(-\frac{3}{2}\) and square it.
\({(\frac{1}{2}(-\frac{3}{2}))}^{2}=\frac{9}{16}\)A mathematical equation demonstrating the process of completing the square. It shows x squared minus three-halves x, plus a blank term, which is indicated by one-half times negative three-halves, all squared, equals 10.
Add \(\frac{9}{16}\) to both sides.A mathematical equation showing x squared minus three-halves x plus nine-sixteenths equals ten plus nine-sixteenths, with the fractions and plus signs in red.
Factor the perfect square trinomial,
writing it as a binomial squared.		A mathematical equation shows '(x - 3/4)^2 = 160/16 + 9/16' written in black characters on a white background.
Add the fractions on the right side.A mathematical equation shows (x - 3/4)^2 = 169/16.
Use the Square Root Property.A mathematical equation is displayed, showing x minus three-fourths equals plus or minus the square root of 169 over 16.
Simplify the radical.A mathematical equation shows 'x minus 3/4 equals plus or minus 13/4' on a white background. This equation represents a step in solving for x, likely from a quadratic equation.
Solve for x.A mathematical equation shows x equals three-fourths plus or minus thirteen-fourths, represented as x = 3/4 ×1 13/4.
Rewrite to show two solutions.Two mathematical equations are displayed horizontally. The first reads x equals 3 over 4 plus 13 over 4. The second reads x equals 3 over 4 minus 13 over 4.
Simplify.The image displays two values for x: x=4 and x=-5/2.
Check:
We leave the check for you!
Try It #17

Solve by completing the square: \(3{r}^{2}-2r=21.\)

\(r=-\frac{7}{3},r=3\)

Did you get it?
Try It #18

Solve by completing the square: \(4{t}^{2}+2t=20.\)

\(t=-\frac{5}{2},t=2\)

Did you get it?

Now that we have seen that the coefficient of x2 must be 1 for us to complete the square, we update our procedure for solving a quadratic equation by completing the square to include equations of the form ax2 + bx + c = 0.

Solve a quadratic equation of the form \(a{x}^{2}+bx+c=0\) by completing the square.
  • Divide by \(a\) to make the coefficient of x2 term 1.
  • Isolate the variable terms on one side and the constant terms on the other.
  • Find \({(\frac{1}{2}\,·\,b)}^{2},\) the number needed to complete the square. Add it to both sides of the equation.
  • Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right
  • Use the Square Root Property.
  • Simplify the radical and then solve the two resulting equations.
  • Check the solutions.
Example 10

Solve by completing the square: \(3{x}^{2}+2x=4.\)

Divide both sides by the coefficient of x² first so you can complete the square as in earlier examples.

Again, our first step will be to make the coefficient of x2 one. By dividing both sides of the equation by the coefficient of x2, we can then continue with solving the equation by completing the square.

Table 12
A mathematical equation reads '3x² + 2x = 4' against a plain white background.
Divide both sides by 3 to make the
coefficient of \({x}^{2}\) equal 1.		A mathematical equation shows (3x^2 + 2x) divided by 3, equaling 4/3. The equation is presented in black text on a white background.
Simplify.A quadratic equation displayed on a white background, reading x squared plus two-thirds x equals four-thirds.
Take half of \(\frac{2}{3}\) and square it.
\({(\frac{1}{2}·\frac{2}{3})}^{2}=\frac{1}{9}\)A quadratic equation, x squared plus two-thirds x plus a blank equals four-thirds. A red box highlights the term one-half times two-thirds, all squared, which is being added to complete the square.
Add \(\frac{1}{9}\) to both sides.A quadratic equation displayed as x squared plus two-thirds x plus one-ninth equals four-thirds plus one-ninth, with the one-ninth terms highlighted in red on both sides.
Factor the perfect square trinomial, writing it as
a binomial squared.		A mathematical equation shows (x + 1/3) squared equals 12/9 plus 1/9, demonstrating steps in solving a quadratic equation or simplifying expressions.
Use the Square Root Property.A mathematical equation is displayed, showing x plus one-third equals plus or minus the square root of thirteen over nine.
Simplify the radical.A mathematical equation is displayed, reading 'x + 1/3 = +/- sqrt(13)/3' on a white background. The equation involves a variable 'x', fractions, a square root, and the plus-minus symbol.
Solve for x .A mathematical equation shows 'X = -1/3 ×1 (sqrt)13 / 3' written in the center of a plain white background.
Rewrite to show two solutions.The image displays two mathematical solutions for x: X = -1/3 + sqrt(13)/3 and X = -1/3 - sqrt(13)/3, representing the roots of a quadratic equation.
Check:
We leave the check for you!
Try It #19

Solve by completing the square: \(4{x}^{2}+3x=2.\)

\(x=-\frac{3}{8}+\frac{\sqrt{41}}{8},\,\,\,x=-\frac{3}{8}-\frac{\sqrt{41}}{8}\)

Did you get it?
Try It #20

Solve by completing the square: \(3{y}^{2}-10y=-5.\)

\(y=\frac{5}{3}+\frac{\sqrt{10}}{3},\,\,\,y=\frac{5}{3}-\frac{\sqrt{10}}{3}\)

Did you get it?

Key Concepts

Section Exercises

Practice Makes Perfect

Complete the Square of a Binomial Expression

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

4

ⓐ \({m}^{2}-24m\) ⓑ \({x}^{2}-11x\) ⓒ \({p}^{2}-\frac{1}{3}p\)

ⓐ \({(m-12)}^{2}\) ⓑ \({(x-\frac{11}{2})}^{2}\)
ⓒ \({(p-\frac{1}{6})}^{2}\)

5

ⓐ \({n}^{2}-16n\) ⓑ \({y}^{2}+15y\) ⓒ \({q}^{2}+\frac{3}{4}q\)

6

ⓐ \({p}^{2}-22p\) ⓑ \({y}^{2}+5y\) ⓒ \({m}^{2}+\frac{2}{5}m\)

ⓐ \({(p-11)}^{2}\) ⓑ \({(y+\frac{5}{2})}^{2}\)
ⓒ \({(m+\frac{1}{5})}^{2}\)

7

ⓐ \({q}^{2}-6q\) ⓑ \({x}^{2}-7x\) ⓒ \({n}^{2}-\frac{2}{3}n\)

Solve Quadratic Equations of the form x2 + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

8

\({u}^{2}+2u=3\)

\(u=-3,u=1\)

9

\({z}^{2}+12z=-11\)

10

\({x}^{2}-20x=21\)

\(x=-1,x=21\)

11

\({y}^{2}-2y=8\)

12

\({m}^{2}+4m=-44\)

\(m=-2\pm 2\sqrt{10}i\)

13

\({n}^{2}-2n=-3\)

14

\({r}^{2}+6r=-11\)

\(r=-3\pm \sqrt{2}i\)

15

\({t}^{2}-14t=-50\)

16

\({a}^{2}-10a=-5\)

\(a=5\pm 2\sqrt{5}\)

17

\({b}^{2}+6b=41\)

18

\({x}^{2}+5x=2\)

\(x=-\frac{5}{2}\pm \frac{\sqrt{33}}{2}\)

19

\({y}^{2}-3y=2\)

20

\({u}^{2}-14u+12=-1\)

\(u=1,u=13\)

21

\({z}^{2}+2z-5=2\)

22

\({r}^{2}-4r-3=9\)

\(r=-2,r=6\)

23

\({t}^{2}-10t-6=5\)

24

\({v}^{2}=9v+2\)

\(v=\frac{9}{2}\pm \frac{\sqrt{89}}{2}\)

25

\({w}^{2}=5w-1\)

26

\({x}^{2}-5=10x\)

\(x=5\pm \sqrt{30}\)

27

\({y}^{2}-14=6y\)

28

\((x+6)(x-2)=9\)

\(x=-7,x=3\)

29

\((y+9)(y+7)=80\)

30

\((x+2)(x+4)=3\)

\(x=-5,x=-1\)

31

\((x-2)(x-6)=5\)

Solve Quadratic Equations of the form ax2 + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

32

\(3{m}^{2}+30m-27=6\)

\(m=-11,m=1\)

33

\(2{x}^{2}-14x+12=0\)

34

\(2{n}^{2}+4n=26\)

\(n=-1\pm \sqrt{14}\)

35

\(5{x}^{2}+20x=15\)

36

\(2{c}^{2}+c=6\)

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

37

\(3{d}^{2}-4d=15\)

38

\(2{x}^{2}+7x-15=0\)

\(x=-5,x=\frac{3}{2}\)

39

\(3{x}^{2}-14x+8=0\)

40

\(2{p}^{2}+7p=14\)

\(p=-\frac{7}{4}\pm \frac{\sqrt{161}}{4}\)

41

\(3{q}^{2}-5q=9\)

42

\(5{x}^{2}-3x=-10\)

\(x=\frac{3}{10}\pm \frac{\sqrt{191}}{10}i\)

43

\(7{x}^{2}+4x=-3\)

Writing Exercises

44

Solve the equation \({x}^{2}+10x=-25\)

ⓐ by using the Square Root Property

ⓑ by Completing the Square

ⓒ Which method do you prefer? Why?

Answers will vary.

45

Solve the equation \({y}^{2}+8y=48\) by completing the square and explain all your steps.

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 complete the square of a binomial expression.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve quadratic equations of the form x squared plus b times x plus c equals 0 by completing the square.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve quadratic equations of the form a times x squared plus b times x plus c equals 0 by completing the square.” “Confidently,” “with some help,” or “No, I don’t get it.”

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?