Pre-Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Subtract $x$ from both sides of the equation.

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

Step 1.3

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

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

Step 1.4

Subtract $x$ from $- 6 x$ .

$3 x^{2} - 7 x = 2$

Step 2

Subtract $2$ from both sides of the equation.

$3 x^{2} - 7 x - 2 = 0$

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 $a = 3$ , $b = - 7$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (3 \cdot - 2)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot - 2}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 2$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot - 2}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 2$ .

$x = \frac{7 \pm \sqrt{49 + 24}}{2 \cdot 3}$

Step 5.1.3

Add $49$ and $24$ .

$x = \frac{7 \pm \sqrt{73}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{7 \pm \sqrt{73}}{6}$

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 $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot - 2}}{2 \cdot 3}$

Step 6.1.2

Multiply $- 4 \cdot 3 \cdot - 2$ .

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot - 2}}{2 \cdot 3}$

Step 6.1.2.2

Multiply $- 12$ by $- 2$ .

$x = \frac{7 \pm \sqrt{49 + 24}}{2 \cdot 3}$

Step 6.1.3

Add $49$ and $24$ .

$x = \frac{7 \pm \sqrt{73}}{2 \cdot 3}$

Step 6.2

Multiply $2$ by $3$ .

$x = \frac{7 \pm \sqrt{73}}{6}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{73}}{6}$

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 $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 3 \cdot - 2}}{2 \cdot 3}$

Step 7.1.2

Multiply $- 4 \cdot 3 \cdot - 2$ .

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Step 7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{7 \pm \sqrt{49 - 12 \cdot - 2}}{2 \cdot 3}$

Step 7.1.2.2

Multiply $- 12$ by $- 2$ .

$x = \frac{7 \pm \sqrt{49 + 24}}{2 \cdot 3}$

Step 7.1.3

Add $49$ and $24$ .

$x = \frac{7 \pm \sqrt{73}}{2 \cdot 3}$

Step 7.2

Multiply $2$ by $3$ .

$x = \frac{7 \pm \sqrt{73}}{6}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{73}}{6}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{73}}{6}, \frac{7 - \sqrt{73}}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 + \sqrt{73}}{6}, \frac{7 - \sqrt{73}}{6}$

Decimal Form:

$x = 2.59066729 \dots, - 0.25733395 \dots$