Algebra Examples

$x^{2} + p x + q = (x - 13) (x + 1)$

Step 1

Simplify $(x - 13) (x + 1)$ .

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Step 1.1

Rewrite.

$x^{2} + p x + q = 0 + 0 + (x - 13) (x + 1)$

Step 1.2

Simplify by adding zeros.

$x^{2} + p x + q = (x - 13) (x + 1)$

Step 1.3

Expand $(x - 13) (x + 1)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$x^{2} + p x + q = x (x + 1) - 13 (x + 1)$

Step 1.3.2

Apply the distributive property.

$x^{2} + p x + q = x \cdot x + x \cdot 1 - 13 (x + 1)$

Step 1.3.3

Apply the distributive property.

$x^{2} + p x + q = x \cdot x + x \cdot 1 - 13 x - 13 \cdot 1$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

$x^{2} + p x + q = x^{2} + x \cdot 1 - 13 x - 13 \cdot 1$

Step 1.4.1.2

Multiply $x$ by $1$ .

$x^{2} + p x + q = x^{2} + x - 13 x - 13 \cdot 1$

Step 1.4.1.3

Multiply $- 13$ by $1$ .

$x^{2} + p x + q = x^{2} + x - 13 x - 13$

Step 1.4.2

Subtract $13 x$ from $x$ .

$x^{2} + p x + q = x^{2} - 12 x - 13$

Step 2

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

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + p x + q - x^{2} = - 12 x - 13$

Step 2.2

Add $12 x$ to both sides of the equation.

$x^{2} + p x + q - x^{2} + 12 x = - 13$

Step 2.3

Combine the opposite terms in $x^{2} + p x + q - x^{2} + 12 x$ .

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Step 2.3.1

Subtract $x^{2}$ from $x^{2}$ .

$p x + q + 0 + 12 x = - 13$

Step 2.3.2

Add $p x + q$ and $0$ .

$p x + q + 12 x = - 13$

Step 3

Subtract $q$ from both sides of the equation.

$p x + 12 x = - 13 - q$

Step 4

Factor $x$ out of $p x + 12 x$ .

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Step 4.1

Factor $x$ out of $p x$ .

$x p + 12 x = - 13 - q$

Step 4.2

Factor $x$ out of $12 x$ .

$x p + x \cdot 12 = - 13 - q$

Step 4.3

Factor $x$ out of $x p + x \cdot 12$ .

$x (p + 12) = - 13 - q$

Step 5

Divide each term in $x (p + 12) = - 13 - q$ by $p + 12$ and simplify.

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Step 5.1

Divide each term in $x (p + 12) = - 13 - q$ by $p + 12$ .

$\frac{x (p + 12)}{p + 12} = \frac{- 13}{p + 12} + \frac{- q}{p + 12}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $p + 12$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{x (p + 12)}{p + 12} = \frac{- 13}{p + 12} + \frac{- q}{p + 12}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 13}{p + 12} + \frac{- q}{p + 12}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move the negative in front of the fraction.

$x = - \frac{13}{p + 12} + \frac{- q}{p + 12}$

Step 5.3.1.2

Move the negative in front of the fraction.

$x = - \frac{13}{p + 12} - \frac{q}{p + 12}$

Step 5.3.2

Simplify terms.

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Step 5.3.2.1

Combine the numerators over the common denominator.

$x = \frac{- 13 - q}{p + 12}$

Step 5.3.2.2

Rewrite $- 13$ as $- 1 (13)$ .

$x = \frac{- 1 (13) - q}{p + 12}$

Step 5.3.2.3

Factor $- 1$ out of $- q$ .

$x = \frac{- 1 (13) - (q)}{p + 12}$

Step 5.3.2.4

Factor $- 1$ out of $- 1 (13) - (q)$ .

$x = \frac{- 1 (13 + q)}{p + 12}$

Step 5.3.2.5

Move the negative in front of the fraction.

$x = - \frac{13 + q}{p + 12}$