# 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.

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

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

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

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

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

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

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

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

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

${n}^{4}-81$

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

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

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

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

${b}^{3}-64$

${m}^{3}+125$

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

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

Use the Zero Product Property

In the following exercises, solve.

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

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

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

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

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

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

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.

Aziza is solving this equation-2(1+x)=4x+10
No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
.473
Angelita
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Angelita
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
p-2/3=5/6 how do I solve it with explanation pls
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
yes
James
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have? Rhonda Reply Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 . Gabriel Reply is that a negative 4 or positive 4? Felix y = mx + b Felix if negative -4, then -4=8/9(0) + b Felix -4=b Felix if positive 4, then 4=b Felix then plug in y=8/9x - 4 or y=8/9x+4 Felix Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost$9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost$6.50 per pound to make?
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up Makaila Reply$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
hola saben como aser un valor de la expresión
NAILEA
integer greater than 2 and less than 12
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy

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