Algebra Examples

$x^{2} + {(x + 4)}^{2} = 20^{2}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the right side.

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Step 1.1.1

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

$x^{2} + {(x + 4)}^{2} = 400$

Step 1.2

Subtract $400$ from both sides of the equation.

$x^{2} + {(x + 4)}^{2} - 400 = 0$

Step 1.3

Simplify $x^{2} + {(x + 4)}^{2} - 400$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$x^{2} + (x + 4) (x + 4) - 400 = 0$

Step 1.3.1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 1.3.1.2.1

Apply the distributive property.

$x^{2} + x (x + 4) + 4 (x + 4) - 400 = 0$

Step 1.3.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 4 + 4 (x + 4) - 400 = 0$

Step 1.3.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 - 400 = 0$

Step 1.3.1.3

Simplify and combine like terms.

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Step 1.3.1.3.1

Simplify each term.

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Step 1.3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 - 400 = 0$

Step 1.3.1.3.1.2

Move $4$ to the left of $x$ .

$x^{2} + x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 - 400 = 0$

Step 1.3.1.3.1.3

Multiply $4$ by $4$ .

$x^{2} + x^{2} + 4 x + 4 x + 16 - 400 = 0$

Step 1.3.1.3.2

Add $4 x$ and $4 x$ .

$x^{2} + x^{2} + 8 x + 16 - 400 = 0$

Step 1.3.2

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

$2 x^{2} + 8 x + 16 - 400 = 0$

Step 1.3.3

Subtract $400$ from $16$ .

$2 x^{2} + 8 x - 384 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = 8$ , and $c = - 384$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (2 \cdot - 384)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 2 \cdot - 384}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot - 384$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 8 \cdot - 384}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $- 384$ .

$x = \frac{- 8 \pm \sqrt{64 + 3072}}{2 \cdot 2}$

Step 4.1.3

Add $64$ and $3072$ .

$x = \frac{- 8 \pm \sqrt{3136}}{2 \cdot 2}$

Step 4.1.4

Rewrite $3136$ as $56^{2}$ .

$x = \frac{- 8 \pm \sqrt{56^{2}}}{2 \cdot 2}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 8 \pm 56}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{- 8 \pm 56}{4}$

Step 4.3

Simplify $\frac{- 8 \pm 56}{4}$ .

$x = - 2 \pm 14$

Step 5

The final answer is the combination of both solutions.

$x = 12, - 16$