# 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

He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least $2,400? Alicia Reply at least 20 Ayla what are the steps? Alicia 6.4 jobs Grahame 32 Grahame what is algebra Azhar Reply repeated addition and subtraction of the order of operations. i love algebra I'm obsessed. Shemiah hi Krekar One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag? Leanna Reply rectangular field solutions Navin Reply What is this? Donna the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is anas Reply ? Choli a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190 Usman Stella bought a dinette set on sale for$725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount? Manhwa Reply 44.19% Scott 40.22% Terence 44.2% Orlando I don't know Donna if you want the discounted price subtract$725 from $1299. then divide the answer by$1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
Can you explain further
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was$1,585. What is the amount of each loan?
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found$7(+7). Totally you lost just $12 (-12) Annushka I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-\$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed