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

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

Step 1

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

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

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Simplify the expression.

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x + 3 = 4 {(x - 2)}^{1}$

Simplify.

$x + 3 = 4 (x - 2)$

Apply the distributive property.

$x + 3 = 4 x + 4 \cdot - 2$

Multiply $4$ by $- 2$ .

$x + 3 = 4 x - 8$

Step 3

Solve for $x$ .

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

Subtract $4 x$ from both sides of the equation.

$x + 3 - 4 x = - 8$

Subtract $4 x$ from $x$ .

$- 3 x + 3 = - 8$

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

Subtract $3$ from both sides of the equation.

$- 3 x = - 8 - 3$

Subtract $3$ from $- 8$ .

$- 3 x = - 11$

Divide each term in $- 3 x = - 11$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 11$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 11}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 11}{- 3}$

Divide $x$ by $1$ .

$x = \frac{- 11}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{11}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{3}$

Decimal Form:

$x = 3.\overline{6}$

Mixed Number Form:

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

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

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