# 10.2 Solve quadratic equations by completing the square  (Page 4/5)

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

Solve $3{x}^{2}-12x-15=0$ by completing the square.

## Solution

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. Factor out the greatest common factor. Divide both sides by 3 to isolate the trinomial. Simplify. Subtract 5 to get the constant terms on the right. Take half of 4 and square it. $\left(\frac{1}{2}\left(4\right){\right)}^{2}=4$ Add 4 to both sides. Factor the perfect square trinomial as a binomial square. Use the Square Root Property. Solve for x. Rewrite to show 2 solutions. Simplify. Check. Solve $2{m}^{2}+16m-8=0$ by completing the square.

$m=-4±2\sqrt{5}$

Solve $4{n}^{2}-24n-56=8$ by completing the square.

$n=-2,8$

To complete the square, the leading coefficient must be one. 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.

Solve $2{x}^{2}-3x=20$ by completing the square.

## Solution

Again, our first step will be to make the coefficient of ${x}^{2}$ be one. By dividing both sides of the equation by the coefficient of ${x}^{2}$ , we can then continue with solving the equation by completing the square. Divide both sides by 2 to get the coefficient of ${x}^{2}$ to be 1. Simplify. Take half of $-\frac{3}{2}$ and square it. $\left(\frac{1}{2}\left(-\frac{3}{2}\right){\right)}^{2}=\frac{9}{16}$ Add $\frac{9}{16}$ to both sides. Factor the perfect square trinomial as a binomial square. Add the fractions on the right side. Use the Square Root Property. Simplify the radical. Solve for x. Rewrite to show 2 solutions. Simplify. Check. We leave the check for you.

Solve $3{r}^{2}-2r=21$ by completing the square.

$r=-\frac{7}{3},r=3$

Solve $4{t}^{2}+2t=20$ by completing the square.

$t=-\frac{5}{2},t=2$

Solve $3{x}^{2}+2x=4$ by completing the square.

## Solution

Again, our first step will be to make the coefficient of ${x}^{2}$ be one. By dividing both sides of the equation by the coefficient of ${x}^{2}$ , we can then continue with solving the equation by completing the square. Divide both sides by 3 to make the coefficient of ${x}^{2}$ equal 1. Simplify. Take half of $\frac{2}{3}$ and square it. $\left(\frac{1}{2}\cdot \frac{2}{3}{\right)}^{2}=\frac{1}{9}$ Add $\frac{1}{9}$ to both sides. Factor the perfect square trinomial as a binomial square. Use the Square Root Property. Simplify the radical. Solve for x . Rewrite to show 2 solutions. Check. We leave the check for you.

Solve $4{x}^{2}+3x=12$ by completing the square.

$x=-\frac{3}{8}±\frac{\sqrt{201}}{8}$

Solve $5{y}^{2}+3y=10$ by completing the square.

$y=-\frac{3}{10}±\frac{\sqrt{209}}{10}$

Access these online resources for additional instruction and practice with solving quadratic equations by completing the square:

## Key concepts

• Binomial Squares Pattern If $a,b$ are real numbers,
${\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}$ ${\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}$ • Complete a Square
To complete the square of ${x}^{2}+bx$ :
1. Identify $b$ , the coefficient of $x$ .
2. Find ${\left(\frac{1}{2}b\right)}^{2}$ , the number to complete the square.
3. Add the ${\left(\frac{1}{2}b\right)}^{2}$ to ${x}^{2}+bx$ .

## 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.

${a}^{2}+10a$

${\left(a+5\right)}^{2}$

${b}^{2}+12b$

${m}^{2}+18m$

${\left(m+9\right)}^{2}$

${n}^{2}+16n$

${m}^{2}-24m$

${\left(m-12\right)}^{2}$

${n}^{2}-16n$

${p}^{2}-22p$

${\left(p-11\right)}^{2}$

${q}^{2}-6q$

${x}^{2}-9x$

${\left(x-\frac{9}{2}\right)}^{2}$

${y}^{2}+11y$

${p}^{2}-\frac{1}{3}p$

${\left(p-\frac{1}{6}\right)}^{2}$

${q}^{2}+\frac{3}{4}q$

Solve Quadratic Equations of the Form ${x}^{2}+bx+c=0$ by Completing the Square

In the following exercises, solve by completing the square.

${v}^{2}+6v=40$

$v=-10,v=4$

${w}^{2}+8w=65$

${u}^{2}+2u=3$

$u=-3,u=1$

${z}^{2}+12z=-11$

${c}^{2}-12c=13$

$c=-1,c=13$

${d}^{2}-8d=9$

${x}^{2}-20x=21$

$x=-1,x=21$

${y}^{2}-2y=8$

${m}^{2}+4m=-44$

no real solution

${n}^{2}-2n=-3$

${r}^{2}+6r=-11$

no real solution

${t}^{2}-14t=-50$

${a}^{2}-10a=-5$

$a=5±2\sqrt{5}$

${b}^{2}+6b=41$

${u}^{2}-14u+12=-1$

$u=1,u=13$

${z}^{2}+2z-5=2$

${v}^{2}=9v+2$

$v=\frac{9}{2}±\frac{\sqrt{89}}{2}$

${w}^{2}=5w-1$

$\left(x+6\right)\left(x-2\right)=9$

$x=-7,x=3$

$\left(y+9\right)\left(y+7\right)=79$

Solve Quadratic Equations of the Form $a{x}^{2}+bx+c=0$ by Completing the Square

In the following exercises, solve by completing the square.

$3{m}^{2}+30m-27=6$

$m=-11,m=1$

$2{n}^{2}+4n-26=0$

$2{c}^{2}+c=6$

$c=-2,c=\frac{3}{2}$

$3{d}^{2}-4d=15$

$2{p}^{2}+7p=14$

$p=-\frac{7}{4}±\frac{\sqrt{161}}{4}$

$3{q}^{2}-5q=9$

## Everyday math

Rafi is designing a rectangular playground to have an area of 320 square feet. He wants one side of the playground to be four feet longer than the other side. Solve the equation ${p}^{2}+4p=320$ for $p$ , the length of one side of the playground. What is the length of the other side?

16 feet, 20 feet

Yvette wants to put a square swimming pool in the corner of her backyard. She will have a 3 foot deck on the south side of the pool and a 9 foot deck on the west side of the pool. She has a total area of 1080 square feet for the pool and two decks. Solve the equation $\left(s+3\right)\left(s+9\right)=1080$ for $s$ , the length of a side of the pool.

## Writing exercises

Solve the equation ${x}^{2}+10x=-25$ by using the Square Root Property and by completing the square. Which method do you prefer? Why?

$-5$ $-5$ Answers will vary.

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

#### Questions & Answers

how to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
What is observation
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him$10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs$9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot? Parker Reply what you wanna get Cesar 800 sq. ft @$1.50 & 200 sq. ft @ $9.00 Marco Geneva treated her parents to dinner at their favorite restaurant. The bill was$74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 =$11.88 total bill: $74.25 +$11.88 = $86.13 ericka yes and tip 16% will be$11.88
David
what is the shorter way to do it
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup? Alexandra Reply Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos Enith Reply June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold? Alexandra Reply I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons ericka Priam has pennies and dimes in a cup holder in his car. The total value of the coins is$4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received$4500 in interest in one year
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Avery and Caden have saved \$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine         By Jordon Humphreys By