Algebra Examples

$x (x + 7) = - 3 (x + 8) - 4$

Step 1

Simplify $x (x + 7)$ .

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Step 1.1

Rewrite.

$0 + 0 + x (x + 7) = - 3 (x + 8) - 4$

Step 1.2

Simplify by adding zeros.

$x (x + 7) = - 3 (x + 8) - 4$

Step 1.3

Apply the distributive property.

$x \cdot x + x \cdot 7 = - 3 (x + 8) - 4$

Step 1.4

Simplify the expression.

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Step 1.4.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 7 = - 3 (x + 8) - 4$

Step 1.4.2

Move $7$ to the left of $x$ .

$x^{2} + 7 x = - 3 (x + 8) - 4$

Step 2

Simplify $- 3 (x + 8) - 4$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$x^{2} + 7 x = - 3 x - 3 \cdot 8 - 4$

Step 2.1.2

Multiply $- 3$ by $8$ .

$x^{2} + 7 x = - 3 x - 24 - 4$

Step 2.2

Subtract $4$ from $- 24$ .

$x^{2} + 7 x = - 3 x - 28$

Step 3

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

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Step 3.1

Add $3 x$ to both sides of the equation.

$x^{2} + 7 x + 3 x = - 28$

Step 3.2

Add $7 x$ and $3 x$ .

$x^{2} + 10 x = - 28$

Step 4

Add $28$ to both sides of the equation.

$x^{2} + 10 x + 28 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = 10$ , and $c = 28$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot 28)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 28}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 28$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 28}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $28$ .

$x = \frac{- 10 \pm \sqrt{100 - 112}}{2 \cdot 1}$

Step 7.1.3

Subtract $112$ from $100$ .

$x = \frac{- 10 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 10 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 10 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 7.1.6

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

$x = \frac{- 10 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 7.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 7.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 10 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 7.1.7.2

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

$x = \frac{- 10 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{- 10 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 7.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 10 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 10 \pm 2 i \sqrt{3}}{2}$

Step 7.3

Simplify $\frac{- 10 \pm 2 i \sqrt{3}}{2}$ .

$x = - 5 \pm i \sqrt{3}$

Step 8

The final answer is the combination of both solutions.

$x = - 5 + i \sqrt{3}, - 5 - i \sqrt{3}$