# 7.6 Quadratic equations  (Page 6/9)

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## Key concepts

• Zero Product Property If $a·b=0$ , then either $a=0$ or $b=0$ or both. See [link] .
• Solve a quadratic equation by factoring To solve a quadratic equation by factoring: See [link] .
1. Write the quadratic equation in standard form, $a{x}^{2}+bx+c=0$ .
3. Use the Zero Product Property.
4. Solve the linear equations.
5. Check.
• Use a problem solving strategy to solve word problems See [link] .
1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

## Practice makes perfect

Use the Zero Product Property

In the following exercises, solve.

$\left(x-3\right)\left(x+7\right)=0$

$x=3,x=-7$

$\left(y-11\right)\left(y+1\right)=0$

$\left(3a-10\right)\left(2a-7\right)=0$

$a=10\text{/}3,a=7\text{/}2$

$\left(5b+1\right)\left(6b+1\right)=0$

$6m\left(12m-5\right)=0$

$m=0,m=5\text{/}12$

$2x\left(6x-3\right)=0$

${\left(y-3\right)}^{2}=0$

$y=3$

${\left(b+10\right)}^{2}=0$

${\left(2x-1\right)}^{2}=0$

$x=1\text{/}2$

${\left(3y+5\right)}^{2}=0$

In the following exercises, solve.

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

$x=3,x=4$ $x=-3,x=-4$

${y}^{2}-8y+15=0$

$5{a}^{2}-26a=24$

$a=-5\text{/}4,a=6$ $a=-4\text{/}5,a=6$

$4{b}^{2}+7b=-3$

$4{m}^{2}=17m-15$

$m=5\text{/}4,m=3$

${n}^{2}=5-6n$ ${n}^{2}=5n-6$

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

$a=-1,a=0$

$12{b}^{2}-15b=-9b$

$49{m}^{2}=144$

$m=12\text{/}7,m=-12\text{/}7$

$625={x}^{2}$

$\left(y-3\right)\left(y+2\right)=4y$

$y=-1,y=6$

$\left(p-5\right)\left(p+3\right)=-7$

$\left(2x+1\right)\left(x-3\right)=-4x$

$x=3\text{/}2,x=-1$

$\left(x+6\right)\left(x-3\right)=-8$

$16{p}^{3}=24{p}^{2}+9p$

$p=0,p=¾$

${m}^{3}-2{m}^{2}=\text{−}m$

$20{x}^{2}-60x=-45$

$x=-2\text{/}3$ $x=3\text{/}2$

$3{y}^{2}-18y=-27$

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive integers is 56. Find the integers.

$7\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8;-8\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-7$

The product of two consecutive integers is 42. Find the integers.

The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.

$4\phantom{\rule{0.2em}{0ex}}\text{feet and}\phantom{\rule{0.2em}{0ex}}7\phantom{\rule{0.2em}{0ex}}\text{feet}$

A rectangular retaining wall has area 15 square feet. The height of the wall is two feet less than its length. Find the height and the length of the wall.

A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.

$6\phantom{\rule{0.2em}{0ex}}\text{feet and}\phantom{\rule{0.2em}{0ex}}8\phantom{\rule{0.2em}{0ex}}\text{feet}$

A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is 9 feet longer than the side along the building. The third side is 7 feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.

Mixed Practice

In the following exercises, solve.

$\left(x+8\right)\left(x-3\right)=0$

$x=-8,x=3$

$\left(3y-5\right)\left(y+7\right)=0$

${p}^{2}+12p+11=0$

$p=-1,p=-11$

${q}^{2}-12q-13=0$

${m}^{2}=6m+16$

$m=-2,m=8$

$4{n}^{2}+19n=5$

${a}^{3}-{a}^{2}-42a=0$

$a=0,a=-6,a=7$

$4{b}^{2}-60b+224=0$

The product of two consecutive integers is 110. Find the integers.

$10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}11;-11\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-10$

The length of one leg of a right triangle is three more than the other leg. If the hypotenuse is 15, find the lengths of the two legs.

## Everyday math

Area of a patio If each side of a square patio is increased by 4 feet, the area of the patio would be 196 square feet. Solve the equation ${\left(s+4\right)}^{2}=196$ for s to find the length of a side of the patio.

10 feet

-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
write in this form a/b answer should be in the simplest form 5%
convert to decimal 9/11
August
Equation in the form of a pending point y+2=1/6(×-4)
write in simplest form 3 4/2
August
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