Algebra Examples

$(x - 1) (x + 5) = - 8$

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 $(x - 1) (x + 5)$ .

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Step 1.1.1.1

Expand $(x - 1) (x + 5)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$x (x + 5) - 1 (x + 5) = - 8$

Step 1.1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 5 - 1 (x + 5) = - 8$

Step 1.1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 5 - 1 x - 1 \cdot 5 = - 8$

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 $x$ .

$x^{2} + x \cdot 5 - 1 x - 1 \cdot 5 = - 8$

Step 1.1.1.2.1.2

Move $5$ to the left of $x$ .

$x^{2} + 5 \cdot x - 1 x - 1 \cdot 5 = - 8$

Step 1.1.1.2.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} + 5 x - x - 1 \cdot 5 = - 8$

Step 1.1.1.2.1.4

Multiply $- 1$ by $5$ .

$x^{2} + 5 x - x - 5 = - 8$

Step 1.1.1.2.2

Subtract $x$ from $5 x$ .

$x^{2} + 4 x - 5 = - 8$

Step 1.2

Add $8$ to both sides of the equation.

$x^{2} + 4 x - 5 + 8 = 0$

Step 1.3

Add $- 5$ and $8$ .

$x^{2} + 4 x + 3 = 0$

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 $a = 1$ , $b = 4$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 4 \pm \sqrt{16 - 12}}{2 \cdot 1}$

Step 4.1.3

Subtract $12$ from $16$ .

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2}{2}$

Step 4.3

Simplify $\frac{- 4 \pm 2}{2}$ .

$x = - 2 \pm 1$

Step 5

The final answer is the combination of both solutions.

$x = - 1, - 3$