Algebra Examples

$(x^{2} + 6 x + 8) (x^{2} + 6 x + 13)$

Step 1

Set $(x^{2} + 6 x + 8) (x^{2} + 6 x + 13)$ equal to $0$ .

$(x^{2} + 6 x + 8) (x^{2} + 6 x + 13) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x^{2} + 6 x + 8 = 0$

$x^{2} + 6 x + 13 = 0$

Step 2.2

Set $x^{2} + 6 x + 8$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $x^{2} + 6 x + 8$ equal to $0$ .

$x^{2} + 6 x + 8 = 0$

Step 2.2.2

Solve $x^{2} + 6 x + 8 = 0$ for $x$ .

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Step 2.2.2.1

Factor $x^{2} + 6 x + 8$ using the AC method.

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Step 2.2.2.1.1

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 $8$ and whose sum is $6$ .

$2, 4$

Step 2.2.2.1.2

Write the factored form using these integers.

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

Step 2.2.2.2

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

$x + 2 = 0$

$x + 4 = 0$

Step 2.2.2.3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.2.2.3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.2.2.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.2.2.4

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

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Step 2.2.2.4.1

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

$x + 4 = 0$

Step 2.2.2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.2.2.5

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

$x = - 2, - 4$

Step 2.3

Set $x^{2} + 6 x + 13$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2} + 6 x + 13$ equal to $0$ .

$x^{2} + 6 x + 13 = 0$

Step 2.3.2

Solve $x^{2} + 6 x + 13 = 0$ for $x$ .

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Step 2.3.2.1

Use the quadratic formula to find the solutions.

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

Step 2.3.2.2

Substitute the values $a = 1$ , $b = 6$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Step 2.3.2.3

Simplify.

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Step 2.3.2.3.1

Simplify the numerator.

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Step 2.3.2.3.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 2.3.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 2.3.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Step 2.3.2.3.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{- 6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Step 2.3.2.3.1.3

Subtract $52$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 2.3.2.3.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 2.3.2.3.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 2.3.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 2.3.2.3.1.7

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

$x = \frac{- 6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 2.3.2.3.1.8

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

$x = \frac{- 6 \pm i \cdot 4}{2 \cdot 1}$

Step 2.3.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 6 \pm 4 i}{2 \cdot 1}$

Step 2.3.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 4 i}{2}$

Step 2.3.2.3.3

Simplify $\frac{- 6 \pm 4 i}{2}$ .

$x = - 3 \pm 2 i$

Step 2.3.2.4

The final answer is the combination of both solutions.

$x = - 3 + 2 i, - 3 - 2 i$

Step 2.4

The final solution is all the values that make $(x^{2} + 6 x + 8) (x^{2} + 6 x + 13) = 0$ true.

$x = - 2, - 4, - 3 + 2 i, - 3 - 2 i$

Step 3