Pre-Algebra Examples

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

Step 1

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

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

Step 2

Add $9$ to both sides of the equation.

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

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 $a = - 1$ , $b = 5$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (- 1 \cdot 9)}}{2 \cdot - 1}$

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 $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot 9}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot 9$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot 9}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $9$ .

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot - 1}$

Step 5.1.3

Add $25$ and $36$ .

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{61}}{- 2}$

Step 5.3

Simplify $\frac{- 5 \pm \sqrt{61}}{- 2}$ .

$x = \frac{5 \pm \sqrt{61}}{2}$

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 $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot 9}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot 9$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot 9}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $9$ .

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot - 1}$

Step 6.1.3

Add $25$ and $36$ .

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{61}}{- 2}$

Step 6.3

Simplify $\frac{- 5 \pm \sqrt{61}}{- 2}$ .

$x = \frac{5 \pm \sqrt{61}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{61}}{2}$

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 $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot 9}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot 9$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot 9}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $9$ .

$x = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot - 1}$

Step 7.1.3

Add $25$ and $36$ .

$x = \frac{- 5 \pm \sqrt{61}}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{61}}{- 2}$

Step 7.3

Simplify $\frac{- 5 \pm \sqrt{61}}{- 2}$ .

$x = \frac{5 \pm \sqrt{61}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{61}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{61}}{2}, \frac{5 - \sqrt{61}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{61}}{2}, \frac{5 - \sqrt{61}}{2}$

Decimal Form:

$x = 6.40512483 \dots, - 1.40512483 \dots$