Question

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find the solution set for each of these open sentences.
(12 points each question)

1. $x^{2}-16=0$
2. $a^{2}-32=4a$
3. $3x^{3}-21x^{2}=-18x$

Explanation:

Step1: Rewrite as difference of squares

$x^2 - 4^2 = 0$

Step2: Factor using difference of squares

$(x-4)(x+4) = 0$

Step3: Solve for x

$x-4=0 \implies x=4$; $x+4=0 \implies x=-4$

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Step1: Rearrange to standard quadratic form

$a^2 -4a -32 = 0$

Step2: Factor the quadratic

$(a-8)(a+4) = 0$

Step3: Solve for a

$a-8=0 \implies a=8$; $a+4=0 \implies a=-4$

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Step1: Rearrange all terms to left side

$3x^3 -21x^2 +18x = 0$

Step2: Factor out common term $3x$

$3x(x^2 -7x +6) = 0$

Step3: Factor the quadratic

$3x(x-6)(x-1) = 0$

Step4: Solve for x

$3x=0 \implies x=0$; $x-6=0 \implies x=6$; $x-1=0 \implies x=1$

Answer:

1. $\{-4, 4\}$
2. $\{-4, 8\}$
3. $\{0, 1, 6\}$