Pre-Algebra Examples

$x^{2} + {(x + 6)}^{2} = 24^{2}$

Step 1

Raise $24$ to the power of $2$ .

$x^{2} + {(x + 6)}^{2} = 576$

Step 2

Subtract $576$ from both sides of the equation.

$x^{2} + {(x + 6)}^{2} - 576 = 0$

Step 3

Simplify $x^{2} + {(x + 6)}^{2} - 576$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$x^{2} + (x + 6) (x + 6) - 576 = 0$

Step 3.1.2

Expand $(x + 6) (x + 6)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$x^{2} + x (x + 6) + 6 (x + 6) - 576 = 0$

Step 3.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 6 + 6 (x + 6) - 576 = 0$

Step 3.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6 - 576 = 0$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 6 + 6 x + 6 \cdot 6 - 576 = 0$

Step 3.1.3.1.2

Move $6$ to the left of $x$ .

$x^{2} + x^{2} + 6 \cdot x + 6 x + 6 \cdot 6 - 576 = 0$

Step 3.1.3.1.3

Multiply $6$ by $6$ .

$x^{2} + x^{2} + 6 x + 6 x + 36 - 576 = 0$

Step 3.1.3.2

Add $6 x$ and $6 x$ .

$x^{2} + x^{2} + 12 x + 36 - 576 = 0$

Step 3.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 12 x + 36 - 576 = 0$

Step 3.3

Subtract $576$ from $36$ .

$2 x^{2} + 12 x - 540 = 0$

Step 4

Factor $2$ out of $2 x^{2} + 12 x - 540$ .

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Step 4.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 12 x - 540 = 0$

Step 4.2

Factor $2$ out of $12 x$ .

$2 (x^{2}) + 2 (6 x) - 540 = 0$

Step 4.3

Factor $2$ out of $- 540$ .

$2 x^{2} + 2 (6 x) + 2 \cdot - 270 = 0$

Step 4.4

Factor $2$ out of $2 x^{2} + 2 (6 x)$ .

$2 (x^{2} + 6 x) + 2 \cdot - 270 = 0$

Step 4.5

Factor $2$ out of $2 (x^{2} + 6 x) + 2 \cdot - 270$ .

$2 (x^{2} + 6 x - 270) = 0$

Step 5

Divide each term in $2 (x^{2} + 6 x - 270) = 0$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 (x^{2} + 6 x - 270) = 0$ by $2$ .

$\frac{2 (x^{2} + 6 x - 270)}{2} = \frac{0}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 6 x - 270)}{2} = \frac{0}{2}$

Step 5.2.1.2

Divide $x^{2} + 6 x - 270$ by $1$ .

$x^{2} + 6 x - 270 = \frac{0}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $2$ .

$x^{2} + 6 x - 270 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = 6$ , and $c = - 270$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 270)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 270}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 270$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 270}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 270$ .

$x = \frac{- 6 \pm \sqrt{36 + 1080}}{2 \cdot 1}$

Step 8.1.3

Add $36$ and $1080$ .

$x = \frac{- 6 \pm \sqrt{1116}}{2 \cdot 1}$

Step 8.1.4

Rewrite $1116$ as $6^{2} \cdot 31$ .

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Step 8.1.4.1

Factor $36$ out of $1116$ .

$x = \frac{- 6 \pm \sqrt{36 (31)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm \sqrt{6^{2} \cdot 31}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 6 \sqrt{31}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 6 \sqrt{31}}{2}$

Step 8.3

Simplify $\frac{- 6 \pm 6 \sqrt{31}}{2}$ .

$x = - 3 \pm 3 \sqrt{31}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 270}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 270$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 270}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 270$ .

$x = \frac{- 6 \pm \sqrt{36 + 1080}}{2 \cdot 1}$

Step 9.1.3

Add $36$ and $1080$ .

$x = \frac{- 6 \pm \sqrt{1116}}{2 \cdot 1}$

Step 9.1.4

Rewrite $1116$ as $6^{2} \cdot 31$ .

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Step 9.1.4.1

Factor $36$ out of $1116$ .

$x = \frac{- 6 \pm \sqrt{36 (31)}}{2 \cdot 1}$

Step 9.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm \sqrt{6^{2} \cdot 31}}{2 \cdot 1}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 6 \sqrt{31}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 6 \sqrt{31}}{2}$

Step 9.3

Simplify $\frac{- 6 \pm 6 \sqrt{31}}{2}$ .

$x = - 3 \pm 3 \sqrt{31}$

Step 9.4

Change the $\pm$ to $+$ .

$x = - 3 + 3 \sqrt{31}$

Step 10

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

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 270}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot - 270$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 270}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $- 270$ .

$x = \frac{- 6 \pm \sqrt{36 + 1080}}{2 \cdot 1}$

Step 10.1.3

Add $36$ and $1080$ .

$x = \frac{- 6 \pm \sqrt{1116}}{2 \cdot 1}$

Step 10.1.4

Rewrite $1116$ as $6^{2} \cdot 31$ .

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Step 10.1.4.1

Factor $36$ out of $1116$ .

$x = \frac{- 6 \pm \sqrt{36 (31)}}{2 \cdot 1}$

Step 10.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm \sqrt{6^{2} \cdot 31}}{2 \cdot 1}$

Step 10.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 6 \sqrt{31}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 6 \sqrt{31}}{2}$

Step 10.3

Simplify $\frac{- 6 \pm 6 \sqrt{31}}{2}$ .

$x = - 3 \pm 3 \sqrt{31}$

Step 10.4

Change the $\pm$ to $-$ .

$x = - 3 - 3 \sqrt{31}$

Step 11

The final answer is the combination of both solutions.

$x = - 3 + 3 \sqrt{31}, - 3 - 3 \sqrt{31}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - 3 + 3 \sqrt{31}, - 3 - 3 \sqrt{31}$

Decimal Form:

$x = 13.70329308 \dots, - 19.70329308 \dots$