Algebra Examples

$x^{3} - 2 x^{2} - 11 x + 12 = x^{3} - 13 x - 12$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $x^{3}$ from both sides of the equation.

$x^{3} - 2 x^{2} - 11 x + 12 - x^{3} = - 13 x - 12$

Step 1.2

Add $13 x$ to both sides of the equation.

$x^{3} - 2 x^{2} - 11 x + 12 - x^{3} + 13 x = - 12$

Step 1.3

Combine the opposite terms in $x^{3} - 2 x^{2} - 11 x + 12 - x^{3} + 13 x$ .

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Step 1.3.1

Subtract $x^{3}$ from $x^{3}$ .

$- 2 x^{2} - 11 x + 12 + 0 + 13 x = - 12$

Step 1.3.2

Add $- 2 x^{2} - 11 x + 12$ and $0$ .

$- 2 x^{2} - 11 x + 12 + 13 x = - 12$

Step 1.4

Add $- 11 x$ and $13 x$ .

$- 2 x^{2} + 2 x + 12 = - 12$

Step 2

Add $12$ to both sides of the equation.

$- 2 x^{2} + 2 x + 12 + 12 = 0$

Step 3

Add $12$ and $12$ .

$- 2 x^{2} + 2 x + 24 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 2$ out of $- 2 x^{2} + 2 x + 24$ .

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Step 4.1.1

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

$- 2 x^{2} + 2 x + 24 = 0$

Step 4.1.2

Factor $- 2$ out of $2 x$ .

$- 2 x^{2} - 2 (- x) + 24 = 0$

Step 4.1.3

Factor $- 2$ out of $24$ .

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

Step 4.1.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- x)$ .

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

Step 4.1.5

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

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

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} - x - 12$ using the AC method.

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Step 4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 4.2.1.2

Write the factored form using these integers.

$- 2 ((x - 4) (x + 3)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- 2 (x - 4) (x + 3) = 0$

Step 5

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

$x - 4 = 0$

$x + 3 = 0$

Step 6

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

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Step 6.1

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

$x - 4 = 0$

Step 6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 7.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 8

The final solution is all the values that make $- 2 (x - 4) (x + 3) = 0$ true.

$x = 4, - 3$

Step 9