Trigonometry Examples

$f (x) = \sqrt{10 x + 7}$ , $f (x) = \sqrt{8 x + 9}$

Step 1

Substitute $\sqrt{8 x + 9}$ for $f (x)$ .

$\sqrt{8 x + 9} = \sqrt{10 x + 7}$

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.

${\sqrt{8 x + 9}}^{2} = {\sqrt{10 x + 7}}^{2}$

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 $\sqrt{8 x + 9}$ as ${(8 x + 9)}^{\frac{1}{2}}$ .

${({(8 x + 9)}^{\frac{1}{2}})}^{2} = {\sqrt{10 x + 7}}^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(8 x + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(8 x + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

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

${(8 x + 9)}^{\frac{1}{2} \cdot 2} = {\sqrt{10 x + 7}}^{2}$

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.

${(8 x + 9)}^{\frac{1}{2} \cdot 2} = {\sqrt{10 x + 7}}^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(8 x + 9)}^{1} = {\sqrt{10 x + 7}}^{2}$

Step 2.2.2.1.2

Simplify.

$8 x + 9 = {\sqrt{10 x + 7}}^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Rewrite ${\sqrt{10 x + 7}}^{2}$ as $10 x + 7$ .

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Step 2.2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 x + 7}$ as ${(10 x + 7)}^{\frac{1}{2}}$ .

$8 x + 9 = {({(10 x + 7)}^{\frac{1}{2}})}^{2}$

Step 2.2.3.1.2

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

$8 x + 9 = {(10 x + 7)}^{\frac{1}{2} \cdot 2}$

Step 2.2.3.1.3

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

$8 x + 9 = {(10 x + 7)}^{\frac{2}{2}}$

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.

$8 x + 9 = {(10 x + 7)}^{\frac{2}{2}}$

Step 2.2.3.1.4.2

Rewrite the expression.

$8 x + 9 = {(10 x + 7)}^{1}$

Step 2.2.3.1.5

Simplify.

$8 x + 9 = 10 x + 7$

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 $10 x$ from both sides of the equation.

$8 x + 9 - 10 x = 7$

Step 2.3.1.2

Subtract $10 x$ from $8 x$ .

$- 2 x + 9 = 7$

Step 2.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.2.1

Subtract $9$ from both sides of the equation.

$- 2 x = 7 - 9$

Step 2.3.2.2

Subtract $9$ from $7$ .

$- 2 x = - 2$

Step 2.3.3

Divide each term in $- 2 x = - 2$ by $- 2$ and simplify.

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Step 2.3.3.1

Divide each term in $- 2 x = - 2$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 2}{- 2}$

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 2}{- 2}$

Step 2.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 2}$

Step 2.3.3.3

Simplify the right side.

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Step 2.3.3.3.1

Divide $- 2$ by $- 2$ .

$x = 1$