Algebra Examples

$(x - 5) (x + 6) = 12$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $(x - 5) (x + 6)$ .

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Step 1.1.1.1

Expand $(x - 5) (x + 6)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$x (x + 6) - 5 (x + 6) = 12$

Step 1.1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 6 - 5 (x + 6) = 12$

Step 1.1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 6 - 5 x - 5 \cdot 6 = 12$

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 6 - 5 x - 5 \cdot 6 = 12$

Step 1.1.1.2.1.2

Move $6$ to the left of $x$ .

$x^{2} + 6 \cdot x - 5 x - 5 \cdot 6 = 12$

Step 1.1.1.2.1.3

Multiply $- 5$ by $6$ .

$x^{2} + 6 x - 5 x - 30 = 12$

Step 1.1.1.2.2

Subtract $5 x$ from $6 x$ .

$x^{2} + x - 30 = 12$

Step 1.2

Subtract $12$ from both sides of the equation.

$x^{2} + x - 30 - 12 = 0$

Step 1.3

Subtract $12$ from $- 30$ .

$x^{2} + x - 42 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = 1$ , and $c = - 42$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 42)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 42}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 42$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 42}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 42$ .

$x = \frac{- 1 \pm \sqrt{1 + 168}}{2 \cdot 1}$

Step 4.1.3

Add $1$ and $168$ .

$x = \frac{- 1 \pm \sqrt{169}}{2 \cdot 1}$

Step 4.1.4

Rewrite $169$ as $13^{2}$ .

$x = \frac{- 1 \pm \sqrt{13^{2}}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 1 \pm 13}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm 13}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 6, - 7$