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Determine the number of solutions to each quadratic equation:

$8{m}^{2}-3m+6=0$ $5{z}^{2}+6z-2=0$ $9{w}^{2}+24w+16=0$ $9{u}^{2}-2u+4=0$

no real solutions 2 1 no real solutions

Determine the number of solutions to each quadratic equation:

${b}^{2}+7b-13=0$ $5{a}^{2}-6a+10=0$ $4{r}^{2}-20r+25=0$ $7{t}^{2}-11t+3=0$

2 no real solutions 1 2

## Identify the most appropriate method to use to solve a quadratic equation

We have used four methods to solve quadratic equations:

• Factoring
• Square Root Property
• Completing the Square

You can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method to use.

## Identify the most appropriate method to solve a quadratic equation.

1. Try Factoring first. If the quadratic factors easily, this method is very quick.
2. Try the Square Root Property next. If the equation fits the form $a{x}^{2}=k$ or $a{\left(x-h\right)}^{2}=k$ , it can easily be solved by using the Square Root Property.
3. Use the Quadratic Formula . Any quadratic equation can be solved by using the Quadratic Formula.

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 this chapter 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 use to solve each quadratic equation:

$5{z}^{2}=17$ $4{x}^{2}-12x+9=0$ $8{u}^{2}+6u=11$

## Solution

$5{z}^{2}=17$

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

$4{x}^{2}-12x+9=0$

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

$8{u}^{2}+6u=11$

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 leads us to choose the Quadratic Formula as the most appropriate method

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

${x}^{2}+6x+8=0$ ${\left(n-3\right)}^{2}=16$ $5{p}^{2}-6p=9$

factor Square Root Property Quadratic Formula

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 Formula factoring Square Root Property

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

## Key concepts

• Quadratic Formula The solutions to a quadratic equation of the form $a{x}^{2}+bx+c=0,$ $a\ne 0$ are given by the formula:
$x=\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}$
1. Write the quadratic formula in standard form. Identify the $a,b,c$ values.
2. Write the quadratic formula. Then substitute in the values of $a,b,c.$
3. Simplify.
4. Check the solutions.
• Using the Discriminant, ${b}^{2}-4ac$ , to Determine the Number of Solutions of a Quadratic Equation
For a quadratic equation of the form $a{x}^{2}+bx+c=0,$ $a\ne 0,$
• if ${b}^{2}-4ac>0$ , the equation has 2 solutions.
• if ${b}^{2}-4ac=0$ , the equation has 1 solution.
• if ${b}^{2}-4ac<0$ , the equation has no real solutions.
• To identify the most appropriate method to solve a quadratic equation:
1. Try Factoring first. If the quadratic factors easily this method is very quick.
2. Try the Square Root Property next. If the equation fits the form $a{x}^{2}=k$ or $a{\left(x-h\right)}^{2}=k$ , it can easily be solved by using the Square Root Property.
3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

## Practice makes perfect

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

$4{m}^{2}+m-3=0$

$m=-1,m=\frac{3}{4}$

$4{n}^{2}-9n+5=0$

$2{p}^{2}-7p+3=0$

$p=\frac{1}{2},p=3$

$3{q}^{2}+8q-3=0$

${p}^{2}+7p+12=0$

$p=-4,p=-3$

${q}^{2}+3q-18=0$

${r}^{2}-8r-33=0$

$r=-3,r=11$

${t}^{2}+13t+40=0$

$3{u}^{2}+7u-2=0$

$u=\frac{-7±\sqrt{73}}{6}$

$6{z}^{2}-9z+1=0$

$2{a}^{2}-6a+3=0$

$a=\frac{3±\sqrt{3}}{2}$

$5{b}^{2}+2b-4=0$

$2{x}^{2}+3x+9=0$

no real solution

$6{y}^{2}-5y+2=0$

$v\left(v+5\right)-10=0$

$v=\frac{-5±\sqrt{65}}{2}$

$3w\left(w-2\right)-8=0$

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

$m=-1,m=\frac{3}{4}$

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

$16{c}^{2}+24c+9=0$

$c=-\frac{3}{4}$

$25{d}^{2}-60d+36=0$

$5{m}^{2}+2m-7=0$

$m=-\frac{7}{5},m=1$

$8{n}^{2}-3n+3=0$

${p}^{2}-6p-27=0$

$p=-3,p=9$

$25{q}^{2}+30q+9=0$

$4{r}^{2}+3r-5=0$

$r=\frac{-3±\sqrt{89}}{8}$

$3t\left(t-2\right)=2$

$2{a}^{2}+12a+5=0$

$a=\frac{-6±\sqrt{26}}{2}$

$4{d}^{2}-7d+2=0$

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

$b=\frac{-2±\sqrt{11}}{6}$

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

$2{x}^{2}+12x-3=0$

$x=\frac{-6±\sqrt{42}}{4}$

$16{y}^{2}+8y+1=0$

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

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

$4{x}^{2}-5x+16=0$
$36{y}^{2}+36y+9=0$
$6{m}^{2}+3m-5=0$
$18{n}^{2}-7n+3=0$

no real solutions 1
2 no real solutions

$9{v}^{2}-15v+25=0$
$100{w}^{2}+60w+9=0$
$5{c}^{2}+7c-10=0$
$15{d}^{2}-4d+8=0$

${r}^{2}+12r+36=0$
$8{t}^{2}-11t+5=0$
$4{u}^{2}-12u+9=0$
$3{v}^{2}-5v-1=0$

1 no real solutions
1 2

$25{p}^{2}+10p+1=0$
$7{q}^{2}-3q-6=0$
$7{y}^{2}+2y+8=0$
$25{z}^{2}-60z+36=0$

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.

${x}^{2}-5x-24=0$
${\left(y+5\right)}^{2}=12$
$14{m}^{2}+3m=11$

factor square root

${\left(8v+3\right)}^{2}=81$
${w}^{2}-9w-22=0$
$4{n}^{2}-10=6$

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

factor square root
factor

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

## Everyday math

A flare is fired straight up from a ship at sea. Solve the equation $16\left({t}^{2}-13t+40\right)=0$ for $t$ , the number of seconds it will take for the flare to be at an altitude of 640 feet.

5 seconds, 8 seconds

An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation $\frac{1}{2}{h}^{2}+3h=140$ for $h$ , the height of the window.

## Writing exercises

Solve the equation ${x}^{2}+10x=200$
by completing the square
Which method do you prefer? Why?

$-20,10$ $-20,10$

Solve the equation $12{y}^{2}+23y=24$
by completing the square
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.

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

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is y=7/5 a solution of 5y+3=10y-4
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