Finite Math Examples

${(x + 7)}^{2} = - 12$

Step 1

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

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Step 1.1

Add $12$ to both sides of the equation.

${(x + 7)}^{2} + 12 = 0$

Step 1.2

Simplify ${(x + 7)}^{2} + 12$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$(x + 7) (x + 7) + 12 = 0$

Step 1.2.1.2

Expand $(x + 7) (x + 7)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$x (x + 7) + 7 (x + 7) + 12 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 7 + 7 (x + 7) + 12 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7 + 12 = 0$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 7 + 7 x + 7 \cdot 7 + 12 = 0$

Step 1.2.1.3.1.2

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

$x^{2} + 7 \cdot x + 7 x + 7 \cdot 7 + 12 = 0$

Step 1.2.1.3.1.3

Multiply $7$ by $7$ .

$x^{2} + 7 x + 7 x + 49 + 12 = 0$

Step 1.2.1.3.2

Add $7 x$ and $7 x$ .

$x^{2} + 14 x + 49 + 12 = 0$

Step 1.2.2

Add $49$ and $12$ .

$x^{2} + 14 x + 61 = 0$

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 $a = 1$ , $b = 14$ , and $c = 61$ into the quadratic formula and solve for $x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (1 \cdot 61)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot 61}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 61$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 61}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $61$ .

$x = \frac{- 14 \pm \sqrt{196 - 244}}{2 \cdot 1}$

Step 4.1.3

Subtract $244$ from $196$ .

$x = \frac{- 14 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 4.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 14 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 4.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 14 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 4.1.6

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

$x = \frac{- 14 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 4.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 4.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 14 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 4.1.7.2

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

$x = \frac{- 14 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{- 14 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 4.1.9

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

$x = \frac{- 14 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- 14 \pm 4 i \sqrt{3}}{2}$

Step 4.3

Simplify $\frac{- 14 \pm 4 i \sqrt{3}}{2}$ .

$x = - 7 \pm 2 i \sqrt{3}$