Trigonometry Examples

$\sqrt{6 x + 3} = 2 \sqrt{x}$

Step 1

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

${\sqrt{6 x + 3}}^{2} = {(2 \sqrt{x})}^{2}$

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 $\sqrt{6 x + 3}$ as ${(6 x + 3)}^{\frac{1}{2}}$ .

${({(6 x + 3)}^{\frac{1}{2}})}^{2} = {(2 \sqrt{x})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(6 x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(6 x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(6 x + 3)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{x})}^{2}$

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.

${(6 x + 3)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{x})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(6 x + 3)}^{1} = {(2 \sqrt{x})}^{2}$

Step 2.2.1.2

Simplify.

$6 x + 3 = {(2 \sqrt{x})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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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 $2 \sqrt{x}$ .

$6 x + 3 = 2^{2} {\sqrt{x}}^{2}$

Step 2.3.1.1.2

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

$6 x + 3 = 4 {\sqrt{x}}^{2}$

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

$6 x + 3 = 4 {(x^{\frac{1}{2}})}^{2}$

Step 2.3.1.2.2

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

$6 x + 3 = 4 x^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

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

$6 x + 3 = 4 x^{\frac{2}{2}}$

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.

$6 x + 3 = 4 x^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$6 x + 3 = 4 x^{1}$

Step 2.3.1.2.5

Simplify.

$6 x + 3 = 4 x$

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 $4 x$ from both sides of the equation.

$6 x + 3 - 4 x = 0$

Step 3.1.2

Subtract $4 x$ from $6 x$ .

$2 x + 3 = 0$

Step 3.2

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 3.3

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 4

Exclude the solutions that do not make $\sqrt{6 x + 3} = 2 \sqrt{x}$ true.

$No solution$