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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 $3{x}^{2}12x15=0$ by completing the square.
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. $\left(\frac{1}{2}\right(4){)}^{2}=4$  
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 $2{m}^{2}+16m8=0$ by completing the square.
$m=\mathrm{4}\pm 2\sqrt{5}$
Solve $4{n}^{2}24n56=8$ by completing the square.
$n=\mathrm{2},8$
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.
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. $\left(\frac{1}{2}\right(\frac{3}{2}){)}^{2}=\frac{9}{16}$  
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.
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. $(\frac{1}{2}\cdot \frac{2}{3}{)}^{2}=\frac{1}{9}$  
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.
$x=\frac{3}{8}\pm \frac{\sqrt{201}}{8}$
Solve $5{y}^{2}+3y=10$ by completing the square.
$y=\frac{3}{10}\pm \frac{\sqrt{209}}{10}$
Access these online resources for additional instruction and practice with solving quadratic equations by completing the square:
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.
${b}^{2}+12b$
${n}^{2}+16n$
${n}^{2}16n$
${q}^{2}6q$
${y}^{2}+11y$
${q}^{2}+\frac{3}{4}q$
Solve Quadratic Equations of the Form ${x}^{2}+bx+c=0$ by Completing the Square
In the following exercises, solve by completing the square.
${w}^{2}+8w=65$
${z}^{2}+12z=\mathrm{11}$
${d}^{2}8d=9$
${y}^{2}2y=8$
${n}^{2}2n=\mathrm{3}$
${t}^{2}14t=\mathrm{50}$
${b}^{2}+6b=41$
${z}^{2}+2z5=2$
${w}^{2}=5w1$
$\left(y+9\right)\left(y+7\right)=79$
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.
$2{n}^{2}+4n26=0$
$3{d}^{2}4d=15$
$3{q}^{2}5q=9$
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.
Solve the equation ${x}^{2}+10x=\mathrm{25}$ ⓐ by using the Square Root Property and ⓑ by completing the square. ⓒ Which method do you prefer? Why?
ⓐ $\mathrm{5}$ ⓑ $\mathrm{5}$ ⓒ Answers will vary.
Solve the equation ${y}^{2}+8y=48$ by completing the square and explain all your steps.
ⓐ 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?
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