Pre-Algebra Examples

$11 x^{2} - 5 x + 14 = 8 + 7 x$

Step 1

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

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Step 1.1

Subtract $7 x$ from both sides of the equation.

$11 x^{2} - 5 x + 14 - 7 x = 8$

Step 1.2

Subtract $7 x$ from $- 5 x$ .

$11 x^{2} - 12 x + 14 = 8$

Step 2

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

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Step 2.1

Subtract $8$ from both sides of the equation.

$11 x^{2} - 12 x + 14 - 8 = 0$

Step 2.2

Subtract $8$ from $14$ .

$11 x^{2} - 12 x + 6 = 0$

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 $a = 11$ , $b = - 12$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (11 \cdot 6)}}{2 \cdot 11}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 11 \cdot 6}}{2 \cdot 11}$

Step 5.1.2

Multiply $- 4 \cdot 11 \cdot 6$ .

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Step 5.1.2.1

Multiply $- 4$ by $11$ .

$x = \frac{12 \pm \sqrt{144 - 44 \cdot 6}}{2 \cdot 11}$

Step 5.1.2.2

Multiply $- 44$ by $6$ .

$x = \frac{12 \pm \sqrt{144 - 264}}{2 \cdot 11}$

Step 5.1.3

Subtract $264$ from $144$ .

$x = \frac{12 \pm \sqrt{- 120}}{2 \cdot 11}$

Step 5.1.4

Rewrite $- 120$ as $- 1 (120)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 120}}{2 \cdot 11}$

Step 5.1.5

Rewrite $\sqrt{- 1 (120)}$ as $\sqrt{- 1} \cdot \sqrt{120}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{120}}{2 \cdot 11}$

Step 5.1.6

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

$x = \frac{12 \pm i \cdot \sqrt{120}}{2 \cdot 11}$

Step 5.1.7

Rewrite $120$ as $2^{2} \cdot 30$ .

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Step 5.1.7.1

Factor $4$ out of $120$ .

$x = \frac{12 \pm i \cdot \sqrt{4 (30)}}{2 \cdot 11}$

Step 5.1.7.2

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2} \cdot 30}}{2 \cdot 11}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (2 \sqrt{30})}{2 \cdot 11}$

Step 5.1.9

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

$x = \frac{12 \pm 2 i \sqrt{30}}{2 \cdot 11}$

Step 5.2

Multiply $2$ by $11$ .

$x = \frac{12 \pm 2 i \sqrt{30}}{22}$

Step 5.3

Simplify $\frac{12 \pm 2 i \sqrt{30}}{22}$ .

$x = \frac{6 \pm i \sqrt{30}}{11}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 11 \cdot 6}}{2 \cdot 11}$

Step 6.1.2

Multiply $- 4 \cdot 11 \cdot 6$ .

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Step 6.1.2.1

Multiply $- 4$ by $11$ .

$x = \frac{12 \pm \sqrt{144 - 44 \cdot 6}}{2 \cdot 11}$

Step 6.1.2.2

Multiply $- 44$ by $6$ .

$x = \frac{12 \pm \sqrt{144 - 264}}{2 \cdot 11}$

Step 6.1.3

Subtract $264$ from $144$ .

$x = \frac{12 \pm \sqrt{- 120}}{2 \cdot 11}$

Step 6.1.4

Rewrite $- 120$ as $- 1 (120)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 120}}{2 \cdot 11}$

Step 6.1.5

Rewrite $\sqrt{- 1 (120)}$ as $\sqrt{- 1} \cdot \sqrt{120}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{120}}{2 \cdot 11}$

Step 6.1.6

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

$x = \frac{12 \pm i \cdot \sqrt{120}}{2 \cdot 11}$

Step 6.1.7

Rewrite $120$ as $2^{2} \cdot 30$ .

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Step 6.1.7.1

Factor $4$ out of $120$ .

$x = \frac{12 \pm i \cdot \sqrt{4 (30)}}{2 \cdot 11}$

Step 6.1.7.2

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2} \cdot 30}}{2 \cdot 11}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (2 \sqrt{30})}{2 \cdot 11}$

Step 6.1.9

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

$x = \frac{12 \pm 2 i \sqrt{30}}{2 \cdot 11}$

Step 6.2

Multiply $2$ by $11$ .

$x = \frac{12 \pm 2 i \sqrt{30}}{22}$

Step 6.3

Simplify $\frac{12 \pm 2 i \sqrt{30}}{22}$ .

$x = \frac{6 \pm i \sqrt{30}}{11}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{6 + i \sqrt{30}}{11}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 11 \cdot 6}}{2 \cdot 11}$

Step 7.1.2

Multiply $- 4 \cdot 11 \cdot 6$ .

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Step 7.1.2.1

Multiply $- 4$ by $11$ .

$x = \frac{12 \pm \sqrt{144 - 44 \cdot 6}}{2 \cdot 11}$

Step 7.1.2.2

Multiply $- 44$ by $6$ .

$x = \frac{12 \pm \sqrt{144 - 264}}{2 \cdot 11}$

Step 7.1.3

Subtract $264$ from $144$ .

$x = \frac{12 \pm \sqrt{- 120}}{2 \cdot 11}$

Step 7.1.4

Rewrite $- 120$ as $- 1 (120)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 120}}{2 \cdot 11}$

Step 7.1.5

Rewrite $\sqrt{- 1 (120)}$ as $\sqrt{- 1} \cdot \sqrt{120}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{120}}{2 \cdot 11}$

Step 7.1.6

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

$x = \frac{12 \pm i \cdot \sqrt{120}}{2 \cdot 11}$

Step 7.1.7

Rewrite $120$ as $2^{2} \cdot 30$ .

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Step 7.1.7.1

Factor $4$ out of $120$ .

$x = \frac{12 \pm i \cdot \sqrt{4 (30)}}{2 \cdot 11}$

Step 7.1.7.2

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

$x = \frac{12 \pm i \cdot \sqrt{2^{2} \cdot 30}}{2 \cdot 11}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (2 \sqrt{30})}{2 \cdot 11}$

Step 7.1.9

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

$x = \frac{12 \pm 2 i \sqrt{30}}{2 \cdot 11}$

Step 7.2

Multiply $2$ by $11$ .

$x = \frac{12 \pm 2 i \sqrt{30}}{22}$

Step 7.3

Simplify $\frac{12 \pm 2 i \sqrt{30}}{22}$ .

$x = \frac{6 \pm i \sqrt{30}}{11}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{6 - i \sqrt{30}}{11}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{6 + i \sqrt{30}}{11}, \frac{6 - i \sqrt{30}}{11}$