Algebra Examples

$12 x^{2} + 48 x = - x^{3} - 64$

Step 1

Add $x^{3}$ to both sides of the equation.

$12 x^{2} + 48 x + x^{3} = - 64$

Step 2

Add $64$ to both sides of the equation.

$12 x^{2} + 48 x + x^{3} + 64 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $12 x$ out of $12 x^{2} + 48 x$ .

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Step 3.1.1

Factor $12 x$ out of $12 x^{2}$ .

$12 x (x) + 48 x + x^{3} + 64 = 0$

Step 3.1.2

Factor $12 x$ out of $48 x$ .

$12 x (x) + 12 x (4) + x^{3} + 64 = 0$

Step 3.1.3

Factor $12 x$ out of $12 x (x) + 12 x (4)$ .

$12 x (x + 4) + x^{3} + 64 = 0$

Step 3.2

Rewrite $64$ as $4^{3}$ .

$12 x (x + 4) + x^{3} + 4^{3} = 0$

Step 3.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 4$ .

$12 x (x + 4) + (x + 4) (x^{2} - x \cdot 4 + 4^{2}) = 0$

Step 3.4

Simplify.

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Step 3.4.1

Multiply $4$ by $- 1$ .

$12 x (x + 4) + (x + 4) (x^{2} - 4 x + 4^{2}) = 0$

Step 3.4.2

Raise $4$ to the power of $2$ .

$12 x (x + 4) + (x + 4) (x^{2} - 4 x + 16) = 0$

Step 3.5

Factor $x + 4$ out of $12 x (x + 4) + (x + 4) (x^{2} - 4 x + 16)$ .

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Step 3.5.1

Factor $x + 4$ out of $12 x (x + 4)$ .

$(x + 4) (12 x) + (x + 4) (x^{2} - 4 x + 16) = 0$

Step 3.5.2

Factor $x + 4$ out of $(x + 4) (12 x) + (x + 4) (x^{2} - 4 x + 16)$ .

$(x + 4) (12 x + x^{2} - 4 x + 16) = 0$

Step 3.6

Subtract $4 x$ from $12 x$ .

$(x + 4) (x^{2} + 8 x + 16) = 0$

Step 3.7

Factor using the perfect square rule.

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Step 3.7.1

Rewrite $16$ as $4^{2}$ .

$(x + 4) (x^{2} + 8 x + 4^{2}) = 0$

Step 3.7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 3.7.3

Rewrite the polynomial.

$(x + 4) (x^{2} + 2 \cdot x \cdot 4 + 4^{2}) = 0$

Step 3.7.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$(x + 4) {(x + 4)}^{2} = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 4 = 0$

${(x + 4)}^{2} = 0$

Step 5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 6

The final solution is all the values that make $(x + 4) {(x + 4)}^{2} = 0$ true.

$x = - 4$