Precalculus Examples

$\sqrt{x + 20} = x$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x + 20}}^{2} = x^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 20}$ as ${(x + 20)}^{\frac{1}{2}}$ .

${({(x + 20)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x + 20)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x + 20)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x + 20)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x + 20)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 20)}^{1} = x^{2}$

Step 2.2.1.2

Simplify.

$x + 20 = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 1$ out of $x + 20 - x^{2}$ .

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Step 3.2.1.1

Reorder the expression.

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Step 3.2.1.1.1

Move $20$ .

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

Step 3.2.1.1.2

Reorder $x$ and $- x^{2}$ .

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

Step 3.2.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + x + 20 = 0$

Step 3.2.1.3

Factor $- 1$ out of $x$ .

$- (x^{2}) - 1 (- x) + 20 = 0$

Step 3.2.1.4

Rewrite $20$ as $- 1 (- 20)$ .

$- (x^{2}) - 1 (- x) - 1 \cdot - 20 = 0$

Step 3.2.1.5

Factor $- 1$ out of $- (x^{2}) - 1 (- x)$ .

$- (x^{2} - x) - 1 \cdot - 20 = 0$

Step 3.2.1.6

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 20)$ .

$- (x^{2} - x - 20) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $x^{2} - x - 20$ using the AC method.

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Step 3.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 3.2.2.1.2

Write the factored form using these integers.

$- ((x - 5) (x + 4)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$- (x - 5) (x + 4) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x + 4 = 0$

Step 3.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 3.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 3.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 3.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.6

The final solution is all the values that make $- (x - 5) (x + 4) = 0$ true.

$x = 5, - 4$

Step 4

Exclude the solutions that do not make $\sqrt{x + 20} = x$ true.

$x = 5$