# 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?

Answers may vary.

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

Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Ericka Reply
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
cricket Reply
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
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
Mikaela Reply
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Sam Reply
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Mckenzie Reply
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Reiley Reply
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
hamzzi Reply
90 minutes
muhammad

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