Solve by Factoring ( square root of 2x)/( square root of x-2)=2

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

Step 1

Subtract $2$ from both sides of the equation.

$\frac{\sqrt{2 x}}{\sqrt{x - 2}} - 2 = 0$

Step 2

Simplify $\frac{\sqrt{2 x}}{\sqrt{x - 2}} - 2$ .

Simplify each term.

Multiply $\frac{\sqrt{2 x}}{\sqrt{x - 2}}$ by $\frac{\sqrt{x - 2}}{\sqrt{x - 2}}$ .

$\frac{\sqrt{2 x}}{\sqrt{x - 2}} \cdot \frac{\sqrt{x - 2}}{\sqrt{x - 2}} - 2 = 0$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2 x}}{\sqrt{x - 2}}$ by $\frac{\sqrt{x - 2}}{\sqrt{x - 2}}$ .

$\frac{\sqrt{2 x} \sqrt{x - 2}}{\sqrt{x - 2} \sqrt{x - 2}} - 2 = 0$

Raise $\sqrt{x - 2}$ to the power of $1$ .

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{\sqrt{x - 2}}^{1} \sqrt{x - 2}} - 2 = 0$

Raise $\sqrt{x - 2}$ to the power of $1$ .

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{\sqrt{x - 2}}^{1} {\sqrt{x - 2}}^{1}} - 2 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{\sqrt{x - 2}}^{1 + 1}} - 2 = 0$

Add $1$ and $1$ .

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{\sqrt{x - 2}}^{2}} - 2 = 0$

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

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

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{({(x - 2)}^{\frac{1}{2}})}^{2}} - 2 = 0$

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

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{(x - 2)}^{\frac{1}{2} \cdot 2}} - 2 = 0$

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

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{(x - 2)}^{\frac{2}{2}}} - 2 = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{(x - 2)}^{\frac{2}{2}}} - 2 = 0$

Rewrite the expression.

$\frac{\sqrt{2 x} \sqrt{x - 2}}{{(x - 2)}^{1}} - 2 = 0$

Simplify.

$\frac{\sqrt{2 x} \sqrt{x - 2}}{x - 2} - 2 = 0$

Combine using the product rule for radicals.

$\frac{\sqrt{2 x (x - 2)}}{x - 2} - 2 = 0$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{\sqrt{2 x (x - 2)}}{x - 2} - 2 \cdot \frac{x - 2}{x - 2} = 0$

Combine $- 2$ and $\frac{x - 2}{x - 2}$ .

$\frac{\sqrt{2 x (x - 2)}}{x - 2} + \frac{- 2 (x - 2)}{x - 2} = 0$

Combine the numerators over the common denominator.

$\frac{\sqrt{2 x (x - 2)} - 2 (x - 2)}{x - 2} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{\sqrt{2 x (x - 2)} - 2 x - 2 \cdot - 2}{x - 2} = 0$

Multiply $- 2$ by $- 2$ .

$\frac{\sqrt{2 x (x - 2)} - 2 x + 4}{x - 2} = 0$

Step 3

Set the numerator equal to zero.

$\sqrt{2 x (x - 2)} - 2 x + 4 = 0$

Step 4

Solve the equation for $x$ .

Move all terms not containing $\sqrt{2 x (x - 2)}$ to the right side of the equation.

Add $2 x$ to both sides of the equation.

$\sqrt{2 x (x - 2)} + 4 = 2 x$

Subtract $4$ from both sides of the equation.

$\sqrt{2 x (x - 2)} = 2 x - 4$

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

${\sqrt{2 x (x - 2)}}^{2} = {(2 x - 4)}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $2$ .

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

Simplify.

$2 x^{2} - 4 x = {(2 x - 4)}^{2}$

Simplify the right side.

Simplify ${(2 x - 4)}^{2}$ .

Rewrite ${(2 x - 4)}^{2}$ as $(2 x - 4) (2 x - 4)$ .

$2 x^{2} - 4 x = (2 x - 4) (2 x - 4)$

Expand $(2 x - 4) (2 x - 4)$ using the FOIL Method.

Apply the distributive property.

$2 x^{2} - 4 x = 2 x (2 x - 4) - 4 (2 x - 4)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 4$ by $2$ .

$2 x^{2} - 4 x = 4 x^{2} - 8 x - 4 (2 x) - 4 \cdot - 4$

Multiply $2$ by $- 4$ .

$2 x^{2} - 4 x = 4 x^{2} - 8 x - 8 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$2 x^{2} - 4 x = 4 x^{2} - 8 x - 8 x + 16$

Subtract $8 x$ from $- 8 x$ .

$2 x^{2} - 4 x = 4 x^{2} - 16 x + 16$

Solve for $x$ .

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

Subtract $4 x^{2}$ from both sides of the equation.

$2 x^{2} - 4 x - 4 x^{2} = - 16 x + 16$

Add $16 x$ to both sides of the equation.

$2 x^{2} - 4 x - 4 x^{2} + 16 x = 16$

Subtract $4 x^{2}$ from $2 x^{2}$ .

$- 2 x^{2} - 4 x + 16 x = 16$

Add $- 4 x$ and $16 x$ .

$- 2 x^{2} + 12 x = 16$

Subtract $16$ from both sides of the equation.

$- 2 x^{2} + 12 x - 16 = 0$

Factor the left side of the equation.

Factor $- 2$ out of $- 2 x^{2} + 12 x - 16$ .

Factor $- 2$ out of $- 2 x^{2}$ .

$- 2 x^{2} + 12 x - 16 = 0$

Factor $- 2$ out of $12 x$ .

$- 2 x^{2} - 2 (- 6 x) - 16 = 0$

Factor $- 2$ out of $- 16$ .

$- 2 x^{2} - 2 (- 6 x) - 2 \cdot 8 = 0$

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 6 x)$ .

$- 2 (x^{2} - 6 x) - 2 \cdot 8 = 0$

Factor $- 2$ out of $- 2 (x^{2} - 6 x) - 2 (8)$ .

$- 2 (x^{2} - 6 x + 8) = 0$

Factor.

Factor $x^{2} - 6 x + 8$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $- 6$ .

$- 4, - 2$

Write the factored form using these integers.

$- 2 ((x - 4) (x - 2)) = 0$

Remove unnecessary parentheses.

$- 2 (x - 4) (x - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x - 2 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $- 2 (x - 4) (x - 2) = 0$ true.

$x = 4, 2$

Step 5

Exclude the solutions that do not make $\frac{\sqrt{2 x (x - 2)} - 2 x + 4}{x - 2} = 0$ true.

$x = 4$

Solve by Factoring ( square root of 2x)/( square root of x-2)=2

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