Solve for x square root of 2x+1=3 square root of x-1

$\sqrt{2 x + 1} = 3 \sqrt{x - 1}$

Step 1

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

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

Step 2

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(3 \sqrt{x - 1})}^{2}$ .

Simplify the expression.

Apply the product rule to $3 \sqrt{x - 1}$ .

$2 x + 1 = 3^{2} {\sqrt{x - 1}}^{2}$

Raise $3$ to the power of $2$ .

$2 x + 1 = 9 {\sqrt{x - 1}}^{2}$

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

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

$2 x + 1 = 9 {({(x - 1)}^{\frac{1}{2}})}^{2}$

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

$2 x + 1 = 9 {(x - 1)}^{\frac{1}{2} \cdot 2}$

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

$2 x + 1 = 9 {(x - 1)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x + 1 = 9 {(x - 1)}^{\frac{2}{2}}$

Rewrite the expression.

$2 x + 1 = 9 {(x - 1)}^{1}$

Simplify.

$2 x + 1 = 9 (x - 1)$

Apply the distributive property.

$2 x + 1 = 9 x + 9 \cdot - 1$

Multiply $9$ by $- 1$ .

$2 x + 1 = 9 x - 9$

Step 3

Solve for $x$ .

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

Subtract $9 x$ from both sides of the equation.

$2 x + 1 - 9 x = - 9$

Subtract $9 x$ from $2 x$ .

$- 7 x + 1 = - 9$

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

Subtract $1$ from both sides of the equation.

$- 7 x = - 9 - 1$

Subtract $1$ from $- 9$ .

$- 7 x = - 10$

Divide each term in $- 7 x = - 10$ by $- 7$ and simplify.

Divide each term in $- 7 x = - 10$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 10}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 10}{- 7}$

Divide $x$ by $1$ .

$x = \frac{- 10}{- 7}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{10}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{10}{7}$

Decimal Form:

$x = 1.\overline{428571}$

Mixed Number Form:

$x = 1 \frac{3}{7}$

Solve for x square root of 2x+1=3 square root of x-1

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