Examples

$| 3 x | - x^{2} = 1$

Step 1

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

$| 3 x | = 1 + x^{2}$

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$3 x = \pm (1 + x^{2})$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$3 x = 1 + x^{2}$

Step 3.2

Subtract $x^{2}$ from both sides of the equation.

$3 x - x^{2} = 1$

Step 3.3

Subtract $1$ from both sides of the equation.

$3 x - x^{2} - 1 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = - 1$ , $b = 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (- 1 \cdot - 1)}}{2 \cdot - 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.6.1.2

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

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Step 3.6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4 \cdot - 1}}{2 \cdot - 1}$

Step 3.6.1.2.2

Multiply $4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4}}{2 \cdot - 1}$

Step 3.6.1.3

Subtract $4$ from $9$ .

$x = \frac{- 3 \pm \sqrt{5}}{2 \cdot - 1}$

Step 3.6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{5}}{- 2}$

Step 3.6.3

Simplify $\frac{- 3 \pm \sqrt{5}}{- 2}$ .

$x = \frac{3 \pm \sqrt{5}}{2}$

Step 3.7

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

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.7.1.2

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

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Step 3.7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4 \cdot - 1}}{2 \cdot - 1}$

Step 3.7.1.2.2

Multiply $4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4}}{2 \cdot - 1}$

Step 3.7.1.3

Subtract $4$ from $9$ .

$x = \frac{- 3 \pm \sqrt{5}}{2 \cdot - 1}$

Step 3.7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{5}}{- 2}$

Step 3.7.3

Simplify $\frac{- 3 \pm \sqrt{5}}{- 2}$ .

$x = \frac{3 \pm \sqrt{5}}{2}$

Step 3.7.4

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{5}}{2}$

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.8.1.2

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

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Step 3.8.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4 \cdot - 1}}{2 \cdot - 1}$

Step 3.8.1.2.2

Multiply $4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4}}{2 \cdot - 1}$

Step 3.8.1.3

Subtract $4$ from $9$ .

$x = \frac{- 3 \pm \sqrt{5}}{2 \cdot - 1}$

Step 3.8.2

Multiply $2$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{5}}{- 2}$

Step 3.8.3

Simplify $\frac{- 3 \pm \sqrt{5}}{- 2}$ .

$x = \frac{3 \pm \sqrt{5}}{2}$

Step 3.8.4

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{5}}{2}$

Step 3.9

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}$

Step 3.10

Next, use the negative value of the $\pm$ to find the second solution.

$3 x = - (1 + x^{2})$

Step 3.11

Simplify $- (1 + x^{2})$ .

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Step 3.11.1

Apply the distributive property.

$3 x = - 1 \cdot 1 - x^{2}$

Step 3.11.2

Multiply $- 1$ by $1$ .

$3 x = - 1 - x^{2}$

Step 3.12

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

$3 x + x^{2} = - 1$

Step 3.13

Add $1$ to both sides of the equation.

$3 x + x^{2} + 1 = 0$

Step 3.14

Use the quadratic formula to find the solutions.

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

Step 3.15

Substitute the values $a = 1$ , $b = 3$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 3.16

Simplify.

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Step 3.16.1

Simplify the numerator.

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Step 3.16.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.16.1.2

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

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Step 3.16.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.16.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Step 3.16.1.3

Subtract $4$ from $9$ .

$x = \frac{- 3 \pm \sqrt{5}}{2 \cdot 1}$

Step 3.16.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{5}}{2}$

Step 3.17

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

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Step 3.17.1

Simplify the numerator.

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Step 3.17.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.17.1.2

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

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Step 3.17.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.17.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Step 3.17.1.3

Subtract $4$ from $9$ .

$x = \frac{- 3 \pm \sqrt{5}}{2 \cdot 1}$

Step 3.17.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{5}}{2}$

Step 3.17.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{5}}{2}$

Step 3.17.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{5}}{2}$

Step 3.17.5

Factor $- 1$ out of $\sqrt{5}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{5})}{2}$

Step 3.17.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{5})$ .

$x = \frac{- 1 (3 - \sqrt{5})}{2}$

Step 3.17.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{5}}{2}$

Step 3.18

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

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Step 3.18.1

Simplify the numerator.

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Step 3.18.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 3.18.1.2

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

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Step 3.18.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Step 3.18.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Step 3.18.1.3

Subtract $4$ from $9$ .

$x = \frac{- 3 \pm \sqrt{5}}{2 \cdot 1}$

Step 3.18.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{5}}{2}$

Step 3.18.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{5}}{2}$

Step 3.18.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{5}}{2}$

Step 3.18.5

Factor $- 1$ out of $- \sqrt{5}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{5})}{2}$

Step 3.18.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{5})$ .

$x = \frac{- 1 (3 + \sqrt{5})}{2}$

Step 3.18.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{5}}{2}$

Step 3.19

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{5}}{2}, - \frac{3 + \sqrt{5}}{2}$

Step 3.20

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}, - \frac{3 - \sqrt{5}}{2}, - \frac{3 + \sqrt{5}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}, - \frac{3 - \sqrt{5}}{2}, - \frac{3 + \sqrt{5}}{2}$

Decimal Form:

$x = 2.61803398 \dots, 0.38196601 \dots, - 0.38196601 \dots, - 2.61803398 \dots$

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