Pre-Algebra Examples

$12 x^{2} + 40 - 7 = (6 x - 1)$

Step 1

Subtract $7$ from $40$ .

$12 x^{2} + 33 = 6 x - 1$

Step 2

Subtract $6 x$ from both sides of the equation.

$12 x^{2} + 33 - 6 x = - 1$

Step 3

Add $1$ to both sides of the equation.

$12 x^{2} + 33 - 6 x + 1 = 0$

Step 4

Add $33$ and $1$ .

$12 x^{2} - 6 x + 34 = 0$

Step 5

Factor $2$ out of $12 x^{2} - 6 x + 34$ .

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Step 5.1

Factor $2$ out of $12 x^{2}$ .

$2 (6 x^{2}) - 6 x + 34 = 0$

Step 5.2

Factor $2$ out of $- 6 x$ .

$2 (6 x^{2}) + 2 (- 3 x) + 34 = 0$

Step 5.3

Factor $2$ out of $34$ .

$2 (6 x^{2}) + 2 (- 3 x) + 2 (17) = 0$

Step 5.4

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

$2 (6 x^{2} - 3 x) + 2 (17) = 0$

Step 5.5

Factor $2$ out of $2 (6 x^{2} - 3 x) + 2 (17)$ .

$2 (6 x^{2} - 3 x + 17) = 0$

Step 6

Divide each term in $2 (6 x^{2} - 3 x + 17) = 0$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 (6 x^{2} - 3 x + 17) = 0$ by $2$ .

$\frac{2 (6 x^{2} - 3 x + 17)}{2} = \frac{0}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 (6 x^{2} - 3 x + 17)}{2} = \frac{0}{2}$

Step 6.2.1.2

Divide $6 x^{2} - 3 x + 17$ by $1$ .

$6 x^{2} - 3 x + 17 = \frac{0}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $2$ .

$6 x^{2} - 3 x + 17 = 0$

Step 7

Use the quadratic formula to find the solutions.

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

Step 8

Substitute the values $a = 6$ , $b = - 3$ , and $c = 17$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (6 \cdot 17)}}{2 \cdot 6}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 6 \cdot 17}}{2 \cdot 6}$

Step 9.1.2

Multiply $- 4 \cdot 6 \cdot 17$ .

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Step 9.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{3 \pm \sqrt{9 - 24 \cdot 17}}{2 \cdot 6}$

Step 9.1.2.2

Multiply $- 24$ by $17$ .

$x = \frac{3 \pm \sqrt{9 - 408}}{2 \cdot 6}$

Step 9.1.3

Subtract $408$ from $9$ .

$x = \frac{3 \pm \sqrt{- 399}}{2 \cdot 6}$

Step 9.1.4

Rewrite $- 399$ as $- 1 (399)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 399}}{2 \cdot 6}$

Step 9.1.5

Rewrite $\sqrt{- 1 (399)}$ as $\sqrt{- 1} \cdot \sqrt{399}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{399}}{2 \cdot 6}$

Step 9.1.6

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

$x = \frac{3 \pm i \sqrt{399}}{2 \cdot 6}$

Step 9.2

Multiply $2$ by $6$ .

$x = \frac{3 \pm i \sqrt{399}}{12}$

Step 10

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

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 6 \cdot 17}}{2 \cdot 6}$

Step 10.1.2

Multiply $- 4 \cdot 6 \cdot 17$ .

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Step 10.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{3 \pm \sqrt{9 - 24 \cdot 17}}{2 \cdot 6}$

Step 10.1.2.2

Multiply $- 24$ by $17$ .

$x = \frac{3 \pm \sqrt{9 - 408}}{2 \cdot 6}$

Step 10.1.3

Subtract $408$ from $9$ .

$x = \frac{3 \pm \sqrt{- 399}}{2 \cdot 6}$

Step 10.1.4

Rewrite $- 399$ as $- 1 (399)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 399}}{2 \cdot 6}$

Step 10.1.5

Rewrite $\sqrt{- 1 (399)}$ as $\sqrt{- 1} \cdot \sqrt{399}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{399}}{2 \cdot 6}$

Step 10.1.6

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

$x = \frac{3 \pm i \sqrt{399}}{2 \cdot 6}$

Step 10.2

Multiply $2$ by $6$ .

$x = \frac{3 \pm i \sqrt{399}}{12}$

Step 10.3

Change the $\pm$ to $+$ .

$x = \frac{3 + i \sqrt{399}}{12}$

Step 11

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

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 6 \cdot 17}}{2 \cdot 6}$

Step 11.1.2

Multiply $- 4 \cdot 6 \cdot 17$ .

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Step 11.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{3 \pm \sqrt{9 - 24 \cdot 17}}{2 \cdot 6}$

Step 11.1.2.2

Multiply $- 24$ by $17$ .

$x = \frac{3 \pm \sqrt{9 - 408}}{2 \cdot 6}$

Step 11.1.3

Subtract $408$ from $9$ .

$x = \frac{3 \pm \sqrt{- 399}}{2 \cdot 6}$

Step 11.1.4

Rewrite $- 399$ as $- 1 (399)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 399}}{2 \cdot 6}$

Step 11.1.5

Rewrite $\sqrt{- 1 (399)}$ as $\sqrt{- 1} \cdot \sqrt{399}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{399}}{2 \cdot 6}$

Step 11.1.6

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

$x = \frac{3 \pm i \sqrt{399}}{2 \cdot 6}$

Step 11.2

Multiply $2$ by $6$ .

$x = \frac{3 \pm i \sqrt{399}}{12}$

Step 11.3

Change the $\pm$ to $-$ .

$x = \frac{3 - i \sqrt{399}}{12}$

Step 12

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{399}}{12}, \frac{3 - i \sqrt{399}}{12}$