Pre-Algebra Examples

$3 x^{2} - x - 2 = - 6 x^{2} + 4 x - 4$

Step 1

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

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Step 1.1

Add $6 x^{2}$ to both sides of the equation.

$3 x^{2} - x - 2 + 6 x^{2} = 4 x - 4$

Step 1.2

Subtract $4 x$ from both sides of the equation.

$3 x^{2} - x - 2 + 6 x^{2} - 4 x = - 4$

Step 1.3

Add $3 x^{2}$ and $6 x^{2}$ .

$9 x^{2} - x - 2 - 4 x = - 4$

Step 1.4

Subtract $4 x$ from $- x$ .

$9 x^{2} - 5 x - 2 = - 4$

Step 2

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

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Step 2.1

Add $4$ to both sides of the equation.

$9 x^{2} - 5 x - 2 + 4 = 0$

Step 2.2

Add $- 2$ and $4$ .

$9 x^{2} - 5 x + 2 = 0$

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 $a = 9$ , $b = - 5$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (9 \cdot 2)}}{2 \cdot 9}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 9 \cdot 2}}{2 \cdot 9}$

Step 5.1.2

Multiply $- 4 \cdot 9 \cdot 2$ .

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Step 5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{5 \pm \sqrt{25 - 36 \cdot 2}}{2 \cdot 9}$

Step 5.1.2.2

Multiply $- 36$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 72}}{2 \cdot 9}$

Step 5.1.3

Subtract $72$ from $25$ .

$x = \frac{5 \pm \sqrt{- 47}}{2 \cdot 9}$

Step 5.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 9}$

Step 5.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 9}$

Step 5.1.6

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

$x = \frac{5 \pm i \sqrt{47}}{2 \cdot 9}$

Step 5.2

Multiply $2$ by $9$ .

$x = \frac{5 \pm i \sqrt{47}}{18}$

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 $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 9 \cdot 2}}{2 \cdot 9}$

Step 6.1.2

Multiply $- 4 \cdot 9 \cdot 2$ .

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Step 6.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{5 \pm \sqrt{25 - 36 \cdot 2}}{2 \cdot 9}$

Step 6.1.2.2

Multiply $- 36$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 72}}{2 \cdot 9}$

Step 6.1.3

Subtract $72$ from $25$ .

$x = \frac{5 \pm \sqrt{- 47}}{2 \cdot 9}$

Step 6.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 9}$

Step 6.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 9}$

Step 6.1.6

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

$x = \frac{5 \pm i \sqrt{47}}{2 \cdot 9}$

Step 6.2

Multiply $2$ by $9$ .

$x = \frac{5 \pm i \sqrt{47}}{18}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{5 + i \sqrt{47}}{18}$

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 $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 9 \cdot 2}}{2 \cdot 9}$

Step 7.1.2

Multiply $- 4 \cdot 9 \cdot 2$ .

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Step 7.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{5 \pm \sqrt{25 - 36 \cdot 2}}{2 \cdot 9}$

Step 7.1.2.2

Multiply $- 36$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 72}}{2 \cdot 9}$

Step 7.1.3

Subtract $72$ from $25$ .

$x = \frac{5 \pm \sqrt{- 47}}{2 \cdot 9}$

Step 7.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 9}$

Step 7.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 9}$

Step 7.1.6

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

$x = \frac{5 \pm i \sqrt{47}}{2 \cdot 9}$

Step 7.2

Multiply $2$ by $9$ .

$x = \frac{5 \pm i \sqrt{47}}{18}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{5 - i \sqrt{47}}{18}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{47}}{18}, \frac{5 - i \sqrt{47}}{18}$