Trigonometry Examples

$\sqrt{5 x - 2} = 2 \sqrt{2 x - 5}$

Step 1

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

${\sqrt{5 x - 2}}^{2} = {(2 \sqrt{2 x - 5})}^{2}$

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 $\sqrt{5 x - 2}$ as ${(5 x - 2)}^{\frac{1}{2}}$ .

${({(5 x - 2)}^{\frac{1}{2}})}^{2} = {(2 \sqrt{2 x - 5})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(5 x - 2)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(5 x - 2)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(5 x - 2)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{2 x - 5})}^{2}$

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.

${(5 x - 2)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{2 x - 5})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(5 x - 2)}^{1} = {(2 \sqrt{2 x - 5})}^{2}$

Step 2.2.1.2

Simplify.

$5 x - 2 = {(2 \sqrt{2 x - 5})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(2 \sqrt{2 x - 5})}^{2}$ .

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Step 2.3.1.1

Simplify the expression.

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Step 2.3.1.1.1

Apply the product rule to $2 \sqrt{2 x - 5}$ .

$5 x - 2 = 2^{2} {\sqrt{2 x - 5}}^{2}$

Step 2.3.1.1.2

Raise $2$ to the power of $2$ .

$5 x - 2 = 4 {\sqrt{2 x - 5}}^{2}$

Step 2.3.1.2

Rewrite ${\sqrt{2 x - 5}}^{2}$ as $2 x - 5$ .

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Step 2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x - 5}$ as ${(2 x - 5)}^{\frac{1}{2}}$ .

$5 x - 2 = 4 {({(2 x - 5)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2.2

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

$5 x - 2 = 4 {(2 x - 5)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

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

$5 x - 2 = 4 {(2 x - 5)}^{\frac{2}{2}}$

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$5 x - 2 = 4 {(2 x - 5)}^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$5 x - 2 = 4 {(2 x - 5)}^{1}$

Step 2.3.1.2.5

Simplify.

$5 x - 2 = 4 (2 x - 5)$

Step 2.3.1.3

Apply the distributive property.

$5 x - 2 = 4 (2 x) + 4 \cdot - 5$

Step 2.3.1.4

Multiply.

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Step 2.3.1.4.1

Multiply $2$ by $4$ .

$5 x - 2 = 8 x + 4 \cdot - 5$

Step 2.3.1.4.2

Multiply $4$ by $- 5$ .

$5 x - 2 = 8 x - 20$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $8 x$ from both sides of the equation.

$5 x - 2 - 8 x = - 20$

Step 3.1.2

Subtract $8 x$ from $5 x$ .

$- 3 x - 2 = - 20$

Step 3.2

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

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Step 3.2.1

Add $2$ to both sides of the equation.

$- 3 x = - 20 + 2$

Step 3.2.2

Add $- 20$ and $2$ .

$- 3 x = - 18$

Step 3.3

Divide each term in $- 3 x = - 18$ by $- 3$ and simplify.

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Step 3.3.1

Divide each term in $- 3 x = - 18$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 18}{- 3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 18}{- 3}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 18}{- 3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 18$ by $- 3$ .

$x = 6$