# 7.6 Quadratic equations  (Page 7/9)

 Page 7 / 9

Watermelon drop A watermelon is dropped from the tenth story of a building. Solve the equation $-16{t}^{2}+144=0$ for $t$ to find the number of seconds it takes the watermelon to reach the ground.

## Writing exercises

Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

## 7.1 Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

$42,60$

6

$450,420$

$90,150,105$

$15$

$60,294,630$

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

$24x-42$

$6\left(4x-7\right)$

$35y+84$

$15{m}^{4}+6{m}^{2}n$

$3{m}^{2}\left(5{m}^{2}+2n\right)$

$24p{t}^{4}+16{t}^{7}$

Factor by Grouping

In the following exercises, factor by grouping.

$ax-ay+bx-by$

$\left(a+b\right)\left(x-y\right)$

${x}^{2}y-x{y}^{2}+2x-2y$

${x}^{2}+7x-3x-21$

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

$4{x}^{2}-16x+3x-12$

${m}^{3}+{m}^{2}+m+1$

$\left({m}^{2}+1\right)\left(m+1\right)$

$5x-5y-y+x$

## 7.2 Factor Trinomials of the form ${x}^{2}+bx+c$

Factor Trinomials of the Form ${x}^{2}+bx+c$

In the following exercises, factor each trinomial of the form ${x}^{2}+bx+c$ .

${u}^{2}+17u+72$

$\left(u+8\right)\left(u+9\right)$

${a}^{2}+14a+33$

${k}^{2}-16k+60$

$\left(k-6\right)\left(k-10\right)$

${r}^{2}-11r+28$

${y}^{2}+6y-7$

$\left(y+7\right)\left(y-1\right)$

${m}^{2}+3m-54$

${s}^{2}-2s-8$

$\left(s-4\right)\left(s+2\right)$

${x}^{2}-3x-10$

Factor Trinomials of the Form ${x}^{2}+bxy+c{y}^{2}$

In the following examples, factor each trinomial of the form ${x}^{2}+bxy+c{y}^{2}$ .

${x}^{2}+12xy+35{y}^{2}$

$\left(x+5y\right)\left(x+7y\right)$

${u}^{2}+14uv+48{v}^{2}$

${a}^{2}+4ab-21{b}^{2}$

$\left(a+7b\right)\left(a-3b\right)$

${p}^{2}-5pq-36{q}^{2}$

## 7.3 Factoring Trinomials of the form $a{x}^{2}+bx+c$

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

${y}^{2}-17y+42$

Undo FOIL

$12{r}^{2}+32r+5$

$8{a}^{3}+72a$

Factor the GCF

$4m-mn-3n+12$

Factor Trinomials of the Form $a{x}^{2}+bx+c$ with a GCF

In the following exercises, factor completely.

$6{x}^{2}+42x+60$

$6\left(x+2\right)\left(x+5\right)$

$8{a}^{2}+32a+24$

$3{n}^{4}-12{n}^{3}-96{n}^{2}$

$3{n}^{2}\left(n-8\right)\left(n+4\right)$

$5{y}^{4}+25{y}^{2}-70y$

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

$2{x}^{2}+9x+4$

$\left(x+4\right)\left(2x+1\right)$

$3{y}^{2}+17y+10$

$18{a}^{2}-9a+1$

$\left(3a-1\right)\left(6a-1\right)$

$8{u}^{2}-14u+3$

$15{p}^{2}+2p-8$

$\left(5p+4\right)\left(3p-2\right)$

$15{x}^{2}+6x-2$

$40{s}^{2}-s-6$

$\left(5s-2\right)\left(8s+3\right)$

$20{n}^{2}-7n-3$

Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

$3{x}^{2}+3x-36$

$3\left(x+4\right)\left(x-3\right)$

$4{x}^{2}+4x-8$

$60{y}^{2}-85y-25$

$5\left(4y+1\right)\left(3y-5\right)$

$18{a}^{2}-57a-21$

## 7.4 Factoring Special Products

Factor Perfect Square Trinomials

In the following exercises, factor.

$25{x}^{2}+30x+9$

${\left(5x+3\right)}^{2}$

$16{y}^{2}+72y+81$

$36{a}^{2}-84ab+49{b}^{2}$

${\left(6a-7b\right)}^{2}$

$64{r}^{2}-176rs+121{s}^{2}$

$40{x}^{2}+360x+810$

$10{\left(2x+9\right)}^{2}$

$75{u}^{2}+180u+108$

$2{y}^{3}-16{y}^{2}+32y$

$2y{\left(y-4\right)}^{2}$

$5{k}^{3}-70{k}^{2}+245k$

Factor Differences of Squares

In the following exercises, factor.

$81{r}^{2}-25$

$\left(9r-5\right)\left(9r+5\right)$

$49{a}^{2}-144$

$169{m}^{2}-{n}^{2}$

$\left(13m+n\right)\left(13m-n\right)$

$64{x}^{2}-{y}^{2}$

$25{p}^{2}-1$

$\left(5p-1\right)\left(5p+1\right)$

$1-16{s}^{2}$

$9-121{y}^{2}$

$\left(3+11y\right)\left(3-11y\right)$

$100{k}^{2}-81$

$20{x}^{2}-125$

$5\left(2x-5\right)\left(2x+5\right)$

$18{y}^{2}-98$

$49{u}^{3}-9u$

$u\left(7u+3\right)\left(7u-3\right)$

$169{n}^{3}-n$

Factor Sums and Differences of Cubes

In the following exercises, factor.

${a}^{3}-125$

$\left(a-5\right)\left({a}^{2}+5a+25\right)$

${b}^{3}-216$

$2{m}^{3}+54$

$2\left(m+3\right)\left({m}^{2}-3m+9\right)$

$81{x}^{3}+3$

## 7.5 General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

$24{x}^{3}+44{x}^{2}$

$4{x}^{2}\left(6x+11\right)$

$24{a}^{4}-9{a}^{3}$

$16{n}^{2}-56mn+49{m}^{2}$

${\left(4n-7m\right)}^{2}$

$6{a}^{2}-25a-9$

$5{r}^{2}+22r-48$

$\left(r+6\right)\left(5r-8\right)$

$5{u}^{4}-45{u}^{2}$

${n}^{4}-81$

$\left({n}^{2}+9\right)\left(n+3\right)\left(n-3\right)$

$64{j}^{2}+225$

$5{x}^{2}+5x-60$

$5\left(x-3\right)\left(x+4\right)$

${b}^{3}-64$

${m}^{3}+125$

$\left(m+5\right)\left({m}^{2}-5m+25\right)$

$2{b}^{2}-2bc+5cb-5{c}^{2}$

## 7.6 Quadratic Equations

Use the Zero Product Property

In the following exercises, solve.

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

$a=3\phantom{\rule{0.2em}{0ex}}a=-7$

$\left(b-3\right)\left(b+10\right)=0$

$3m\left(2m-5\right)\left(m+6\right)=0$

$m=0\phantom{\rule{0.2em}{0ex}}m=-3\phantom{\rule{0.2em}{0ex}}m=\frac{5}{2}$

$7n\left(3n+8\right)\left(n-5\right)=0$

Solve Quadratic Equations by Factoring

In the following exercises, solve.

${x}^{2}+9x+20=0$

$x=-4,x=-5$

${y}^{2}-y-72=0$

$2{p}^{2}-11p=40$

$p=-\frac{5}{2},p=8$

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

$144{m}^{2}-25=0$

$m=\frac{5}{12},m=-\frac{5}{12}$

$4{n}^{2}=36$

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive numbers is $462$ . Find the numbers.

$-21,-22\phantom{\rule{0.2em}{0ex}}21,22$

The area of a rectangular shaped patio $400$ square feet. The length of the patio is $9$ feet more than its width. Find the length and width.

## Practice test

In the following exercises, find the Greatest Common Factor in each expression.

$14y-42$

$7\left(y-6\right)$

$-6{x}^{2}-30x$

$80{a}^{2}+120{a}^{3}$

$40{a}^{2}\left(2+3a\right)$

$5m\left(m-1\right)+3\left(m-1\right)$

In the following exercises, factor completely.

${x}^{2}+13x+36$

$\left(x+7\right)\left(x+6\right)$

${p}^{2}+pq-12{q}^{2}$

$3{a}^{3}-6{a}^{2}-72a$

$3a\left({a}^{2}-2a-14\right)$

${s}^{2}-25s+84$

$5{n}^{2}+30n+45$

$5\left(n+1\right)\left(n+5\right)$

$64{y}^{2}-49$

$xy-8y+7x-56$

$\left(x-8\right)\left(y+7\right)$

$40{r}^{2}+810$

$9{s}^{2}-12s+4$

${\left(3s-2\right)}^{2}$

${n}^{2}+12n+36$

$100-{a}^{2}$

$\left(10-a\right)\left(10+a\right)$

$6{x}^{2}-11x-10$

$3{x}^{2}-75{y}^{2}$

$3\left(x+5y\right)\left(x-5y\right)$

${c}^{3}-1000{d}^{3}$

$ab-3b-2a+6$

$\left(a-3\right)\left(b-2\right)$

$6{u}^{2}+3u-18$

$8{m}^{2}+22m+5$

$\left(4m+1\right)\left(2m+5\right)$

In the following exercises, solve.

${x}^{2}+9x+20=0$

${y}^{2}=y+132$

$\text{y}=-11,\text{y}=12$

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

$9{b}^{2}-9=0$

$b=1,b=-1$

$16-{m}^{2}=0$

$4{n}^{2}+19+21=0$

$n=-\frac{7}{4},n=-3$

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

The product of two consecutive integers is $156$ . Find the integers.

$12\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}13;-13\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-12$

The area of a rectangular place mat is $168$ square inches. Its length is two inches longer than the width. Find the length and width of the placemat.

#### Questions & Answers

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
6.25
Ciid
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Ciid
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
is anyone else having issues with the links not doing anything?
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Enter the answers in decimal form.
Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at$4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use? Samer 4x6.25=$25 coffee blend 4×4.40= $17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds$77.96 they will spend in total
tyler
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.
i need help how to do this is confusing
what kind of math is it?
Danteii
help me to understand
huh, what is the algebra problem
Daniel
How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers?
tyler
What is the domain and range of heaviside
What is the domain and range of Heaviside and signum
Christopher
25-35
Fazal
The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Tickets for a show are $70 for adults and$50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled \$17,200. How many adult tickets and how many child tickets were sold?
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?   By Lakeima Roberts      By By