Question

solve each equation by factoring

1. $x^2 = 2x$
2. $x^2 = 18 + 3x$
3. $x^2 - 5x = 6$
4. $n^2 + 12n = -35$
5. $k^2 + 7k = 8$
6. $n^2 - 14n = -48$

Explanation:

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Problem 15: $x^2 = 2x$

Step1: Rearrange to standard form

$x^2 - 2x = 0$

Step2: Factor out common term

$x(x - 2) = 0$

Step3: Apply zero product rule

$x=0$ or $x-2=0$
$x=0$ or $x=2$

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Problem 16: $x^2 = 18 + 3x$

Step1: Rearrange to standard form

$x^2 - 3x - 18 = 0$

Step2: Factor the quadratic

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

Step3: Apply zero product rule

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

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Problem 17: $x^2 - 5x = 6$

Step1: Rearrange to standard form

$x^2 - 5x - 6 = 0$

Step2: Factor the quadratic

$(x - 6)(x + 1) = 0$

Step3: Apply zero product rule

$x-6=0$ or $x+1=0$
$x=6$ or $x=-1$

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Problem 18: $n^2 + 12n = -35$

Step1: Rearrange to standard form

$n^2 + 12n + 35 = 0$

Step2: Factor the quadratic

$(n + 5)(n + 7) = 0$

Step3: Apply zero product rule

$n+5=0$ or $n+7=0$
$n=-5$ or $n=-7$

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Problem 19: $k^2 + 7k = 8$

Step1: Rearrange to standard form

$k^2 + 7k - 8 = 0$

Step2: Factor the quadratic

$(k + 8)(k - 1) = 0$

Step3: Apply zero product rule

$k+8=0$ or $k-1=0$
$k=-8$ or $k=1$

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Problem 20: $n^2 - 14n = -48$

Step1: Rearrange to standard form

$n^2 - 14n + 48 = 0$

Step2: Factor the quadratic

$(n - 6)(n - 8) = 0$

Step3: Apply zero product rule

$n-6=0$ or $n-8=0$
$n=6$ or $n=8$

Answer:

1. $x=0$ or $x=2$
2. $x=6$ or $x=-3$
3. $x=6$ or $x=-1$
4. $n=-5$ or $n=-7$
5. $k=-8$ or $k=1$
6. $n=6$ or $n=8$