Question

name__________
date________ period________
solving quadratic equations by factoring
solve each equation by factoring.

1. $(k + 1)(k - 5) = 0$
2. $(a + 1)(a + 2) = 0$
3. $(4k + 5)(k + 1) = 0$
4. $(2m + 3)(4m + 3) = 0$
5. $x^2 - 11x + 19 = -5$
6. $n^2 + 7n + 15 = 5$
7. $n^2 - 10n + 22 = -2$
8. $n^2 + 3n - 12 = 6$
9. $6n^2 - 18n - 18 = 6$
10. $7r^2 - 14r = -7$

Explanation:

1) Step1: Apply zero product property

Set each factor equal to 0:
$k+1=0$ or $k-5=0$

1) Step2: Solve for $k$

$k=-1$ or $k=5$

2) Step1: Apply zero product property

Set each factor equal to 0:
$a+1=0$ or $a+2=0$

2) Step2: Solve for $a$

$a=-1$ or $a=-2$

3) Step1: Apply zero product property

Set each factor equal to 0:
$4k+5=0$ or $k+1=0$

3) Step2: Solve for $k$

$k=-\frac{5}{4}$ or $k=-1$

4) Step1: Apply zero product property

Set each factor equal to 0:
$2m+3=0$ or $4m+3=0$

4) Step2: Solve for $m$

$m=-\frac{3}{2}$ or $m=-\frac{3}{4}$

5) Step1: Rewrite in standard form

Add 5 to both sides:
$x^2-11x+24=0$

5) Step2: Factor the quadratic

$(x-3)(x-8)=0$

5) Step3: Apply zero product property

$x-3=0$ or $x-8=0$

5) Step4: Solve for $x$

$x=3$ or $x=8$

6) Step1: Rewrite in standard form

Subtract 5 from both sides:
$n^2+7n+10=0$

6) Step2: Factor the quadratic

$(n+2)(n+5)=0$

6) Step3: Apply zero product property

$n+2=0$ or $n+5=0$

6) Step4: Solve for $n$

$n=-2$ or $n=-5$

7) Step1: Rewrite in standard form

Add 2 to both sides:
$n^2-10n+24=0$

7) Step2: Factor the quadratic

$(n-4)(n-6)=0$

7) Step3: Apply zero product property

$n-4=0$ or $n-6=0$

7) Step4: Solve for $n$

$n=4$ or $n=6$

8) Step1: Rewrite in standard form

Subtract 6 from both sides:
$n^2+3n-18=0$

8) Step2: Factor the quadratic

$(n+6)(n-3)=0$

8) Step3: Apply zero product property

$n+6=0$ or $n-3=0$

8) Step4: Solve for $n$

$n=-6$ or $n=3$

9) Step1: Rewrite in standard form

Subtract 6 from both sides:
$6n^2-18n-24=0$

9) Step2: Simplify the equation

Divide by 6:
$n^2-3n-4=0$

9) Step3: Factor the quadratic

$(n-4)(n+1)=0$

9) Step4: Apply zero product property

$n-4=0$ or $n+1=0$

9) Step5: Solve for $n$

$n=4$ or $n=-1$

10) Step1: Rewrite in standard form

Add 7 to both sides:
$7r^2-14r+7=0$

10) Step2: Simplify the equation

Divide by 7:
$r^2-2r+1=0$

10) Step3: Factor the quadratic

$(r-1)^2=0$

10) Step4: Solve for $r$

$r=1$ (repeated root)

Answer:

1. $k=-1$ or $k=5$
2. $a=-1$ or $a=-2$
3. $k=-\frac{5}{4}$ or $k=-1$
4. $m=-\frac{3}{2}$ or $m=-\frac{3}{4}$
5. $x=3$ or $x=8$
6. $n=-2$ or $n=-5$
7. $n=4$ or $n=6$
8. $n=-6$ or $n=3$
9. $n=4$ or $n=-1$
10. $r=1$