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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?

Equation in the form of a pending point y+2=1/6(×-4)
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her$66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
x > $110,000 bruce greater than$110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost$4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use? nawal Reply$1.38 worth of strawberries + $1.14 worth of blueberries which=$2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times$2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60 Melissa good Robinson 👍 Leitha thanks Melissa. Leitha nawal let's do another😊 Leitha we can't use emojis...I see now Leitha Sorry for the multi post. My phone glitches. Melissa Vina has$4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs$3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound? Vonna Reply 33.336 Robinson Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost$8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be$10 per square foot?
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have? mikayla Reply 2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75 Melissa John has$175 in $5 and$10 bills in his drawer. The number of $5 bills is three times the number of$10 bills. How many of each are in the drawer?
7-$10 21-$5
Robert
Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the$4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total \$23,500? (Round your answer to the nearest tenth of a percent.)
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert
what is the answer for this 3×9+28÷4-8
315
lashonna
how do you do xsquard+7x+10=0
What
(x + 2)(x + 5), then set each factor to zero and solve for x. so, x = -2 and x = -5.
bruce
I skipped it
What
In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil