Trigonometry Examples

$f (x) = \sqrt{x + 1}$ , $f (x) = \sqrt{x - 6}$

Step 1

Substitute $\sqrt{x - 6}$ for $f (x)$ .

$\sqrt{x - 6} = \sqrt{x + 1}$

Step 2

Solve for $x$ .

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Step 2.1

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

${\sqrt{x - 6}}^{2} = {\sqrt{x + 1}}^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 6}$ as ${(x - 6)}^{\frac{1}{2}}$ .

${({(x - 6)}^{\frac{1}{2}})}^{2} = {\sqrt{x + 1}}^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(x - 6)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(x - 6)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

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

${(x - 6)}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 1}}^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(x - 6)}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 1}}^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(x - 6)}^{1} = {\sqrt{x + 1}}^{2}$

Step 2.2.2.1.2

Simplify.

$x - 6 = {\sqrt{x + 1}}^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

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Step 2.2.3.1.1

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

$x - 6 = {({(x + 1)}^{\frac{1}{2}})}^{2}$

Step 2.2.3.1.2

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

$x - 6 = {(x + 1)}^{\frac{1}{2} \cdot 2}$

Step 2.2.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$x - 6 = {(x + 1)}^{\frac{2}{2}}$

Step 2.2.3.1.4

Cancel the common factor of $2$ .

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Step 2.2.3.1.4.1

Cancel the common factor.

$x - 6 = {(x + 1)}^{\frac{2}{2}}$

Step 2.2.3.1.4.2

Rewrite the expression.

$x - 6 = {(x + 1)}^{1}$

Step 2.2.3.1.5

Simplify.

$x - 6 = x + 1$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.1.1

Subtract $x$ from both sides of the equation.

$x - 6 - x = 1$

Step 2.3.1.2

Combine the opposite terms in $x - 6 - x$ .

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Step 2.3.1.2.1

Subtract $x$ from $x$ .

$0 - 6 = 1$

Step 2.3.1.2.2

Subtract $6$ from $0$ .

$- 6 = 1$

Step 2.3.2

Since $- 6 \neq 1$ , there are no solutions.

No solution