Trigonometry Examples

$x = \sqrt{4 y + 12}$

Step 1

Rewrite the equation as $\sqrt{4 y + 12} = x$ .

$\sqrt{4 y + 12} = x$

Step 2

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

${\sqrt{4 y + 12}}^{2} = x^{2}$

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 $\sqrt{4 y + 12}$ as ${(4 y + 12)}^{\frac{1}{2}}$ .

${({(4 y + 12)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(4 y + 12)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(4 y + 12)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(4 y + 12)}^{\frac{1}{2} \cdot 2} = x^{2}$

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.

${(4 y + 12)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 y + 12)}^{1} = x^{2}$

Step 3.2.1.2

Simplify.

$4 y + 12 = x^{2}$

Step 4

Solve for $y$ .

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Step 4.1

Subtract $12$ from both sides of the equation.

$4 y = x^{2} - 12$

Step 4.2

Divide each term in $4 y = x^{2} - 12$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 y = x^{2} - 12$ by $4$ .

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{- 12}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{- 12}{4}$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{4} + \frac{- 12}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 12$ by $4$ .

$y = \frac{x^{2}}{4} - 3$

Step 5

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 6