Pre-Algebra Examples

$x^{2} + 1 x = 5 x^{2} + 5 x - 5$

Step 1

Multiply $x$ by $1$ .

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

Step 2

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

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Step 2.1

Subtract $5 x^{2}$ from both sides of the equation.

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

Step 2.2

Subtract $5 x$ from both sides of the equation.

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

Step 2.3

Subtract $5 x^{2}$ from $x^{2}$ .

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

Step 2.4

Subtract $5 x$ from $x$ .

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

Step 3

Add $5$ to both sides of the equation.

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

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 4$ , $b = - 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4 \cdot 5}}{2 \cdot - 4}$

Step 6.1.2

Multiply $- 4 \cdot - 4 \cdot 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16 \cdot 5}}{2 \cdot - 4}$

Step 6.1.2.2

Multiply $16$ by $5$ .

$x = \frac{4 \pm \sqrt{16 + 80}}{2 \cdot - 4}$

Step 6.1.3

Add $16$ and $80$ .

$x = \frac{4 \pm \sqrt{96}}{2 \cdot - 4}$

Step 6.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 6.1.4.1

Factor $16$ out of $96$ .

$x = \frac{4 \pm \sqrt{16 (6)}}{2 \cdot - 4}$

Step 6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot - 4}$

Step 6.1.5

Pull terms out from under the radical.

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

Step 6.2

Multiply $2$ by $- 4$ .

$x = \frac{4 \pm 4 \sqrt{6}}{- 8}$

Step 6.3

Simplify $\frac{4 \pm 4 \sqrt{6}}{- 8}$ .

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

Step 6.4

Move the negative in front of the fraction.

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

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 $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4 \cdot 5}}{2 \cdot - 4}$

Step 7.1.2

Multiply $- 4 \cdot - 4 \cdot 5$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16 \cdot 5}}{2 \cdot - 4}$

Step 7.1.2.2

Multiply $16$ by $5$ .

$x = \frac{4 \pm \sqrt{16 + 80}}{2 \cdot - 4}$

Step 7.1.3

Add $16$ and $80$ .

$x = \frac{4 \pm \sqrt{96}}{2 \cdot - 4}$

Step 7.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 7.1.4.1

Factor $16$ out of $96$ .

$x = \frac{4 \pm \sqrt{16 (6)}}{2 \cdot - 4}$

Step 7.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot - 4}$

Step 7.1.5

Pull terms out from under the radical.

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

Step 7.2

Multiply $2$ by $- 4$ .

$x = \frac{4 \pm 4 \sqrt{6}}{- 8}$

Step 7.3

Simplify $\frac{4 \pm 4 \sqrt{6}}{- 8}$ .

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

Step 7.4

Move the negative in front of the fraction.

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

Step 7.5

Change the $\pm$ to $+$ .

$x = - \frac{1 + \sqrt{6}}{2}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 4 \cdot 5}}{2 \cdot - 4}$

Step 8.1.2

Multiply $- 4 \cdot - 4 \cdot 5$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{4 \pm \sqrt{16 + 16 \cdot 5}}{2 \cdot - 4}$

Step 8.1.2.2

Multiply $16$ by $5$ .

$x = \frac{4 \pm \sqrt{16 + 80}}{2 \cdot - 4}$

Step 8.1.3

Add $16$ and $80$ .

$x = \frac{4 \pm \sqrt{96}}{2 \cdot - 4}$

Step 8.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 8.1.4.1

Factor $16$ out of $96$ .

$x = \frac{4 \pm \sqrt{16 (6)}}{2 \cdot - 4}$

Step 8.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot - 4}$

Step 8.1.5

Pull terms out from under the radical.

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

Step 8.2

Multiply $2$ by $- 4$ .

$x = \frac{4 \pm 4 \sqrt{6}}{- 8}$

Step 8.3

Simplify $\frac{4 \pm 4 \sqrt{6}}{- 8}$ .

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

Step 8.4

Move the negative in front of the fraction.

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

Step 8.5

Change the $\pm$ to $-$ .

$x = - \frac{1 - \sqrt{6}}{2}$

Step 9

The final answer is the combination of both solutions.

$x = - \frac{1 + \sqrt{6}}{2}, - \frac{1 - \sqrt{6}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 + \sqrt{6}}{2}, - \frac{1 - \sqrt{6}}{2}$

Decimal Form:

$x = - 1.72474487 \dots, 0.72474487 \dots$