Pre-Algebra Examples

$(x) (4 x + 9) = 520$

Step 1

Simplify $x (4 x + 9)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x (4 x) + x \cdot 9 = 520$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$4 x \cdot x + x \cdot 9 = 520$

Step 1.1.2.2

Move $9$ to the left of $x$ .

$4 x \cdot x + 9 \cdot x = 520$

Step 1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1

Move $x$ .

$4 (x \cdot x) + 9 \cdot x = 520$

Step 1.2.2

Multiply $x$ by $x$ .

$4 x^{2} + 9 \cdot x = 520$

$4 x^{2} + 9 x = 520$

Step 2

Subtract $520$ from both sides of the equation.

$4 x^{2} + 9 x - 520 = 0$

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 $a = 4$ , $b = 9$ , and $c = - 520$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (4 \cdot - 520)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 4 \cdot - 520}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 520$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 16 \cdot - 520}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 520$ .

$x = \frac{- 9 \pm \sqrt{81 + 8320}}{2 \cdot 4}$

Step 5.1.3

Add $81$ and $8320$ .

$x = \frac{- 9 \pm \sqrt{8401}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{- 9 \pm \sqrt{8401}}{8}$

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 $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 4 \cdot - 520}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 520$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 16 \cdot - 520}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 520$ .

$x = \frac{- 9 \pm \sqrt{81 + 8320}}{2 \cdot 4}$

Step 6.1.3

Add $81$ and $8320$ .

$x = \frac{- 9 \pm \sqrt{8401}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{- 9 \pm \sqrt{8401}}{8}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 9 + \sqrt{8401}}{8}$

Step 6.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 + \sqrt{8401}}{8}$

Step 6.5

Factor $- 1$ out of $\sqrt{8401}$ .

$x = \frac{- 1 \cdot 9 - 1 (- \sqrt{8401})}{8}$

Step 6.6

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{8401})$ .

$x = \frac{- 1 (9 - \sqrt{8401})}{8}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{9 - \sqrt{8401}}{8}$

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 $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 4 \cdot - 520}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 520$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 16 \cdot - 520}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 520$ .

$x = \frac{- 9 \pm \sqrt{81 + 8320}}{2 \cdot 4}$

Step 7.1.3

Add $81$ and $8320$ .

$x = \frac{- 9 \pm \sqrt{8401}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{- 9 \pm \sqrt{8401}}{8}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 9 - \sqrt{8401}}{8}$

Step 7.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 - \sqrt{8401}}{8}$

Step 7.5

Factor $- 1$ out of $- \sqrt{8401}$ .

$x = \frac{- 1 \cdot 9 - (\sqrt{8401})}{8}$

Step 7.6

Factor $- 1$ out of $- 1 (9) - (\sqrt{8401})$ .

$x = \frac{- 1 (9 + \sqrt{8401})}{8}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{9 + \sqrt{8401}}{8}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{9 - \sqrt{8401}}{8}, - \frac{9 + \sqrt{8401}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 - \sqrt{8401}}{8}, - \frac{9 + \sqrt{8401}}{8}$

Decimal Form:

$x = 10.33212114 \dots, - 12.58212114 \dots$