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Watermelon drop A watermelon is dropped from the tenth story of a building. Solve the equation $\mathrm{-16}{t}^{2}+144=0$ for $t$ to find the number of seconds it takes the watermelon to reach the ground.
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.
ⓐ 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?
Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
$450,420$
$60,294,630$
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
$24p{t}^{4}+16{t}^{7}$
Factor by Grouping
In the following exercises, factor by grouping.
${x}^{2}y-x{y}^{2}+2x-2y$
$4{x}^{2}-16x+3x-12$
$5x-5y-y+x$
Factor Trinomials of the Form ${x}^{2}+bx+c$
In the following exercises, factor each trinomial of the form ${x}^{2}+bx+c$ .
${a}^{2}+14a+33$
${r}^{2}-11r+28$
${m}^{2}+3m-54$
${x}^{2}-3x-10$
Factor Trinomials of the Form ${x}^{2}+bxy+c{y}^{2}$
In the following examples, factor each trinomial of the form ${x}^{2}+bxy+c{y}^{2}$ .
${x}^{2}+12xy+35{y}^{2}$
$\left(x+5y\right)\left(x+7y\right)$
${u}^{2}+14uv+48{v}^{2}$
${a}^{2}+4ab-21{b}^{2}$
$\left(a+7b\right)\left(a-3b\right)$
${p}^{2}-5pq-36{q}^{2}$
Recognize a Preliminary Strategy to Factor Polynomials Completely
In the following exercises, identify the best method to use to factor each polynomial.
$12{r}^{2}+32r+5$
$4m-mn-3n+12$
Factor Trinomials of the Form $a{x}^{2}+bx+c$ with a GCF
In the following exercises, factor completely.
$8{a}^{2}+32a+24$
$3{n}^{4}-12{n}^{3}-96{n}^{2}$
$3{n}^{2}\left(n-8\right)\left(n+4\right)$
$5{y}^{4}+25{y}^{2}-70y$
Factor Trinomials Using the “ac” Method
In the following exercises, factor.
$3{y}^{2}+17y+10$
$8{u}^{2}-14u+3$
$15{x}^{2}+6x-2$
$20{n}^{2}-7n-3$
Factor Trinomials with a GCF Using the “ac” Method
In the following exercises, factor.
$4{x}^{2}+4x-8$
$18{a}^{2}-57a-21$
Factor Perfect Square Trinomials
In the following exercises, factor.
$16{y}^{2}+72y+81$
$64{r}^{2}-176rs+121{s}^{2}$
$75{u}^{2}+180u+108$
$5{k}^{3}-70{k}^{2}+245k$
Factor Differences of Squares
In the following exercises, factor.
$49{a}^{2}-144$
$64{x}^{2}-{y}^{2}$
$1-16{s}^{2}$
$100{k}^{2}-81$
$18{y}^{2}-98$
$169{n}^{3}-n$
Factor Sums and Differences of Cubes
In the following exercises, factor.
${b}^{3}-216$
$81{x}^{3}+3$
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
$24{a}^{4}-9{a}^{3}$
$6{a}^{2}-25a-9$
$5{u}^{4}-45{u}^{2}$
${n}^{4}-81$
$\left({n}^{2}+9\right)\left(n+3\right)\left(n-3\right)$
$64{j}^{2}+225$
${b}^{3}-64$
$2{b}^{2}-2bc+5cb-5{c}^{2}$
Use the Zero Product Property
In the following exercises, solve.
$(a-3)(a+7)=0$
$a=3\phantom{\rule{0.2em}{0ex}}a=\mathrm{-7}$
$(b-3)(b+10)=0$
$3m(2m-5)(m+6)=0$
$m=0\phantom{\rule{0.2em}{0ex}}m=\mathrm{-3}\phantom{\rule{0.2em}{0ex}}m=\frac{5}{2}$
$7n(3n+8)(n-5)=0$
Solve Quadratic Equations by Factoring
In the following exercises, solve.
${y}^{2}-y-72=0$
${q}^{3}+3{q}^{2}+2q=0$
$4{n}^{2}=36$
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
The product of two consecutive numbers is $462$ . Find the numbers.
$\mathrm{-21},\mathrm{-22}\phantom{\rule{0.2em}{0ex}}21,22$
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.
In the following exercises, find the Greatest Common Factor in each expression.
$\mathrm{-6}{x}^{2}-30x$
$5m\left(m-1\right)+3\left(m-1\right)$
In the following exercises, factor completely.
${p}^{2}+pq-12{q}^{2}$
${s}^{2}-25s+84$
$64{y}^{2}-49$
$40{r}^{2}+810$
${n}^{2}+12n+36$
$6{x}^{2}-11x-10$
${c}^{3}-1000{d}^{3}$
$6{u}^{2}+3u-18$
In the following exercises, solve.
${x}^{2}+9x+20=0$
$5{a}^{2}+26a=24$
$16-{m}^{2}=0$
$\left(x-3\right)\left(x+2\right)=6$
The product of two consecutive integers is $156$ . Find the integers.
$12\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}13;-13\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\mathrm{-12}$
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.
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