Finite Math Examples

$3 x^{2} + 6 x + 6 = 0$

Step 1

Factor $3$ out of $3 x^{2} + 6 x + 6$ .

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Step 1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 6 x + 6 = 0$

Step 1.2

Factor $3$ out of $6 x$ .

$3 (x^{2}) + 3 (2 x) + 6 = 0$

Step 1.3

Factor $3$ out of $6$ .

$3 x^{2} + 3 (2 x) + 3 \cdot 2 = 0$

Step 1.4

Factor $3$ out of $3 x^{2} + 3 (2 x)$ .

$3 (x^{2} + 2 x) + 3 \cdot 2 = 0$

Step 1.5

Factor $3$ out of $3 (x^{2} + 2 x) + 3 \cdot 2$ .

$3 (x^{2} + 2 x + 2) = 0$

Step 2

Divide each term in $3 (x^{2} + 2 x + 2) = 0$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 (x^{2} + 2 x + 2) = 0$ by $3$ .

$\frac{3 (x^{2} + 2 x + 2)}{3} = \frac{0}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} + 2 x + 2)}{3} = \frac{0}{3}$

Step 2.2.1.2

Divide $x^{2} + 2 x + 2$ by $1$ .

$x^{2} + 2 x + 2 = \frac{0}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $3$ .

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = 2$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 5.1.3

Subtract $8$ from $4$ .

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

Step 5.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 5.1.6

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

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

Step 5.1.7

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

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

Step 5.1.8

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

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

Step 5.1.9

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

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

Step 5.2

Multiply $2$ by $1$ .

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

Step 5.3

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

$x = - 1 \pm i$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 6.1.3

Subtract $8$ from $4$ .

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

Step 6.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 6.1.6

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

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

Step 6.1.7

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

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

Step 6.1.8

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

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

Step 6.1.9

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

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

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

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

$x = - 1 \pm i$

Step 6.4

Change the $\pm$ to $+$ .

$x = - 1 + i$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 7.1.3

Subtract $8$ from $4$ .

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

Step 7.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 7.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 7.1.6

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

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

Step 7.1.7

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

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

Step 7.1.8

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

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

Step 7.1.9

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

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

Step 7.2

Multiply $2$ by $1$ .

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

Step 7.3

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

$x = - 1 \pm i$

Step 7.4

Change the $\pm$ to $-$ .

$x = - 1 - i$

Step 8

The final answer is the combination of both solutions.

$x = - 1 + i, - 1 - i$