Solve by Factoring square root of 4x+1 = square root of x+10

$\sqrt{4 x + 1} = \sqrt{x + 10}$

Step 1

Subtract $\sqrt{x + 10}$ from both sides of the equation.

$\sqrt{4 x + 1} - \sqrt{x + 10} = 0$

Step 2

Add $\sqrt{x + 10}$ to both sides of the equation.

$\sqrt{4 x + 1} = \sqrt{x + 10}$

Step 3

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

${\sqrt{4 x + 1}}^{2} = {\sqrt{x + 10}}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 1}$ as ${(4 x + 1)}^{\frac{1}{2}}$ .

${({(4 x + 1)}^{\frac{1}{2}})}^{2} = {\sqrt{x + 10}}^{2}$

Simplify the left side.

Simplify ${({(4 x + 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 x + 1)}^{\frac{1}{2}})}^{2}$ .

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

${(4 x + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 10}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 10}}^{2}$

Rewrite the expression.

${(4 x + 1)}^{1} = {\sqrt{x + 10}}^{2}$

Simplify.

$4 x + 1 = {\sqrt{x + 10}}^{2}$

Simplify the right side.

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

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

$4 x + 1 = {({(x + 10)}^{\frac{1}{2}})}^{2}$

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

$4 x + 1 = {(x + 10)}^{\frac{1}{2} \cdot 2}$

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

$4 x + 1 = {(x + 10)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x + 1 = {(x + 10)}^{\frac{2}{2}}$

Rewrite the expression.

$4 x + 1 = {(x + 10)}^{1}$

Simplify.

$4 x + 1 = x + 10$

Step 5

Solve for $x$ .

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

Subtract $x$ from both sides of the equation.

$4 x + 1 - x = 10$

Subtract $x$ from $4 x$ .

$3 x + 1 = 10$

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

Subtract $1$ from both sides of the equation.

$3 x = 10 - 1$

Subtract $1$ from $10$ .

$3 x = 9$

Divide each term in $3 x = 9$ by $3$ and simplify.

Divide each term in $3 x = 9$ by $3$ .

$\frac{3 x}{3} = \frac{9}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{9}{3}$

Divide $x$ by $1$ .

$x = \frac{9}{3}$

Simplify the right side.

Divide $9$ by $3$ .

$x = 3$

Solve by Factoring square root of 4x+1 = square root of x+10

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