Solve the Rational Equation for x square root of 6x+5 = square root of 2x+10

$\sqrt{6 x + 5} = \sqrt{2 x + 10}$

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

${\sqrt{6 x + 5}}^{2} = {\sqrt{2 x + 10}}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x + 5}$ as ${(6 x + 5)}^{\frac{1}{2}}$ .

${({(6 x + 5)}^{\frac{1}{2}})}^{2} = {\sqrt{2 x + 10}}^{2}$

Simplify the left side.

Simplify ${({(6 x + 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(6 x + 5)}^{\frac{1}{2}})}^{2}$ .

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

${(6 x + 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{2 x + 10}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(6 x + 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{2 x + 10}}^{2}$

Rewrite the expression.

${(6 x + 5)}^{1} = {\sqrt{2 x + 10}}^{2}$

Simplify.

$6 x + 5 = {\sqrt{2 x + 10}}^{2}$

Simplify the right side.

Simplify ${\sqrt{2 x + 10}}^{2}$ .

Factor $2$ out of $2 x + 10$ .

Factor $2$ out of $2 x$ .

$6 x + 5 = {\sqrt{2 (x) + 10}}^{2}$

Factor $2$ out of $10$ .

$6 x + 5 = {\sqrt{2 x + 2 \cdot 5}}^{2}$

Factor $2$ out of $2 x + 2 \cdot 5$ .

$6 x + 5 = {\sqrt{2 (x + 5)}}^{2}$

Rewrite ${\sqrt{2 (x + 5)}}^{2}$ as $2 (x + 5)$ .

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

$6 x + 5 = {({(2 (x + 5))}^{\frac{1}{2}})}^{2}$

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

$6 x + 5 = {(2 (x + 5))}^{\frac{1}{2} \cdot 2}$

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

$6 x + 5 = {(2 (x + 5))}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$6 x + 5 = {(2 (x + 5))}^{\frac{2}{2}}$

Rewrite the expression.

$6 x + 5 = {(2 (x + 5))}^{1}$

Simplify.

$6 x + 5 = 2 (x + 5)$

Apply the distributive property.

$6 x + 5 = 2 x + 2 \cdot 5$

Multiply $2$ by $5$ .

$6 x + 5 = 2 x + 10$

Solve for $x$ .

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

Subtract $2 x$ from both sides of the equation.

$6 x + 5 - 2 x = 10$

Subtract $2 x$ from $6 x$ .

$4 x + 5 = 10$

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

Subtract $5$ from both sides of the equation.

$4 x = 10 - 5$

Subtract $5$ from $10$ .

$4 x = 5$

Divide each term in $4 x = 5$ by $4$ and simplify.

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Divide $x$ by $1$ .

$x = \frac{5}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{4}$

Decimal Form:

$x = 1.25$

Mixed Number Form:

$x = 1 \frac{1}{4}$

Solve the Rational Equation for x square root of 6x+5 = square root of 2x+10

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