Pre-Algebra Examples

$x^{2} - 20 x = 7 - 83$

Step 1

Subtract $83$ from $7$ .

$x^{2} - 20 x = - 76$

Step 2

Add $76$ to both sides of the equation.

$x^{2} - 20 x + 76 = 0$

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 $a = 1$ , $b = - 20$ , and $c = 76$ into the quadratic formula and solve for $x$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (1 \cdot 76)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 76$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 76}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $76$ .

$x = \frac{20 \pm \sqrt{400 - 304}}{2 \cdot 1}$

Step 5.1.3

Subtract $304$ from $400$ .

$x = \frac{20 \pm \sqrt{96}}{2 \cdot 1}$

Step 5.1.4

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

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Step 5.1.4.1

Factor $16$ out of $96$ .

$x = \frac{20 \pm \sqrt{16 (6)}}{2 \cdot 1}$

Step 5.1.4.2

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

$x = \frac{20 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 4 \sqrt{6}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{20 \pm 4 \sqrt{6}}{2}$

Step 5.3

Simplify $\frac{20 \pm 4 \sqrt{6}}{2}$ .

$x = 10 \pm 2 \sqrt{6}$

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 $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 76$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 76}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $76$ .

$x = \frac{20 \pm \sqrt{400 - 304}}{2 \cdot 1}$

Step 6.1.3

Subtract $304$ from $400$ .

$x = \frac{20 \pm \sqrt{96}}{2 \cdot 1}$

Step 6.1.4

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

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Step 6.1.4.1

Factor $16$ out of $96$ .

$x = \frac{20 \pm \sqrt{16 (6)}}{2 \cdot 1}$

Step 6.1.4.2

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

$x = \frac{20 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 4 \sqrt{6}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{20 \pm 4 \sqrt{6}}{2}$

Step 6.3

Simplify $\frac{20 \pm 4 \sqrt{6}}{2}$ .

$x = 10 \pm 2 \sqrt{6}$

Step 6.4

Change the $\pm$ to $+$ .

$x = 10 + 2 \sqrt{6}$

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 $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 76$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 76}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $76$ .

$x = \frac{20 \pm \sqrt{400 - 304}}{2 \cdot 1}$

Step 7.1.3

Subtract $304$ from $400$ .

$x = \frac{20 \pm \sqrt{96}}{2 \cdot 1}$

Step 7.1.4

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

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Step 7.1.4.1

Factor $16$ out of $96$ .

$x = \frac{20 \pm \sqrt{16 (6)}}{2 \cdot 1}$

Step 7.1.4.2

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

$x = \frac{20 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 4 \sqrt{6}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{20 \pm 4 \sqrt{6}}{2}$

Step 7.3

Simplify $\frac{20 \pm 4 \sqrt{6}}{2}$ .

$x = 10 \pm 2 \sqrt{6}$

Step 7.4

Change the $\pm$ to $-$ .

$x = 10 - 2 \sqrt{6}$

Step 8

The final answer is the combination of both solutions.

$x = 10 + 2 \sqrt{6}, 10 - 2 \sqrt{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 10 + 2 \sqrt{6}, 10 - 2 \sqrt{6}$

Decimal Form:

$x = 14.89897948 \dots, 5.10102051 \dots$