Trigonometry Examples

$x = \sqrt{- 4 x + 32}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{- 4 x + 32} = x$

Step 2

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

${\sqrt{- 4 x + 32}}^{2} = x^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 4 x + 32}$ as ${(- 4 x + 32)}^{\frac{1}{2}}$ .

${({(- 4 x + 32)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(- 4 x + 32)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(- 4 x + 32)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(- 4 x + 32)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(- 4 x + 32)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(- 4 x + 32)}^{1} = x^{2}$

Step 3.2.1.2

Simplify.

$- 4 x + 32 = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$- 4 x + 32 - x^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $- 1$ out of $- 4 x + 32 - x^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $32$ .

$- 4 x - x^{2} + 32 = 0$

Step 4.2.1.1.2

Reorder $- 4 x$ and $- x^{2}$ .

$- x^{2} - 4 x + 32 = 0$

Step 4.2.1.2

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

$- (x^{2}) - 4 x + 32 = 0$

Step 4.2.1.3

Factor $- 1$ out of $- 4 x$ .

$- (x^{2}) - (4 x) + 32 = 0$

Step 4.2.1.4

Rewrite $32$ as $- 1 (- 32)$ .

$- (x^{2}) - (4 x) - 1 \cdot - 32 = 0$

Step 4.2.1.5

Factor $- 1$ out of $- (x^{2}) - (4 x)$ .

$- (x^{2} + 4 x) - 1 \cdot - 32 = 0$

Step 4.2.1.6

Factor $- 1$ out of $- (x^{2} + 4 x) - 1 (- 32)$ .

$- (x^{2} + 4 x - 32) = 0$

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor $x^{2} + 4 x - 32$ using the AC method.

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Step 4.2.2.1.1

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 $- 32$ and whose sum is $4$ .

$- 4, 8$

Step 4.2.2.1.2

Write the factored form using these integers.

$- ((x - 4) (x + 8)) = 0$

Step 4.2.2.2

Remove unnecessary parentheses.

$- (x - 4) (x + 8) = 0$

Step 4.3

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

$x - 4 = 0$

$x + 8 = 0$

Step 4.4

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

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Step 4.4.1

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

$x - 4 = 0$

Step 4.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.5

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 4.5.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 4.6

The final solution is all the values that make $- (x - 4) (x + 8) = 0$ true.

$x = 4, - 8$

Step 5

Exclude the solutions that do not make $x = \sqrt{- 4 x + 32}$ true.

$x = 4$