Solve for x square root of 4x-2 = square root of 2x-2

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

Solve for $\sqrt{4 x}$ .

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

Rewrite.

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

Simplify by adding zeros.

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

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

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 2$ .

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

Factor $2$ out of $2 (x) + 2 (- 1)$ .

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

Add $2$ to both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

Expand $(\sqrt{2 (x - 1)} + 2) (\sqrt{2 (x - 1)} + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{2 (x - 1)} \sqrt{2 (x - 1)}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Move $2$ to the left of $\sqrt{2 (x - 1)}$ .

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

Multiply $2$ by $2$ .

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

Add $- 2$ and $4$ .

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

Add $2 \sqrt{2 (x - 1)}$ and $2 \sqrt{2 (x - 1)}$ .

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

Solve for $4 \sqrt{2 (x - 1)}$ .

Rewrite the equation as $2 x + 2 + 4 \sqrt{2 (x - 1)} = 4 x$ .

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

Move all terms not containing $4 \sqrt{2 (x - 1)}$ to the right side of the equation.

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

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

Subtract $2$ from both sides of the equation.

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

Subtract $2 x$ from $4 x$ .

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the distributive property.

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

Simplify the expression.

Multiply $2$ by $- 1$ .

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

Apply the product rule to $4 {(2 x - 2)}^{\frac{1}{2}}$ .

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

Raise $4$ to the power of $2$ .

$16 {({(2 x - 2)}^{\frac{1}{2}})}^{2} = {(2 x - 2)}^{2}$

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

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

$16 {(2 x - 2)}^{\frac{1}{2} \cdot 2} = {(2 x - 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(2 x - 2)}^{\frac{1}{2} \cdot 2} = {(2 x - 2)}^{2}$

Rewrite the expression.

$16 {(2 x - 2)}^{1} = {(2 x - 2)}^{2}$

Simplify.

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

Apply the distributive property.

$16 (2 x) + 16 \cdot - 2 = {(2 x - 2)}^{2}$

Multiply.

Multiply $2$ by $16$ .

$32 x + 16 \cdot - 2 = {(2 x - 2)}^{2}$

Multiply $16$ by $- 2$ .

$32 x - 32 = {(2 x - 2)}^{2}$

Simplify the right side.

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

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

$32 x - 32 = (2 x - 2) (2 x - 2)$

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

Apply the distributive property.

$32 x - 32 = 2 x (2 x - 2) - 2 (2 x - 2)$

Apply the distributive property.

$32 x - 32 = 2 x (2 x) + 2 x \cdot - 2 - 2 (2 x - 2)$

Apply the distributive property.

$32 x - 32 = 2 x (2 x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$32 x - 32 = 2 \cdot 2 x \cdot x + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

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

Move $x$ .

$32 x - 32 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Multiply $x$ by $x$ .

$32 x - 32 = 2 \cdot 2 x^{2} + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Multiply $2$ by $2$ .

$32 x - 32 = 4 x^{2} + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2$

Multiply $- 2$ by $2$ .

$32 x - 32 = 4 x^{2} - 4 x - 2 (2 x) - 2 \cdot - 2$

Multiply $2$ by $- 2$ .

$32 x - 32 = 4 x^{2} - 4 x - 4 x - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$32 x - 32 = 4 x^{2} - 4 x - 4 x + 4$

Subtract $4 x$ from $- 4 x$ .

$32 x - 32 = 4 x^{2} - 8 x + 4$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} - 8 x + 4 = 32 x - 32$

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

Subtract $32 x$ from both sides of the equation.

$4 x^{2} - 8 x + 4 - 32 x = - 32$

Subtract $32 x$ from $- 8 x$ .

$4 x^{2} - 40 x + 4 = - 32$

Add $32$ to both sides of the equation.

$4 x^{2} - 40 x + 4 + 32 = 0$

Add $4$ and $32$ .

$4 x^{2} - 40 x + 36 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 40 x + 36$ .

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 40 x + 36 = 0$

Factor $4$ out of $- 40 x$ .

$4 (x^{2}) + 4 (- 10 x) + 36 = 0$

Factor $4$ out of $36$ .

$4 x^{2} + 4 (- 10 x) + 4 \cdot 9 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 10 x)$ .

$4 (x^{2} - 10 x) + 4 \cdot 9 = 0$

Factor $4$ out of $4 (x^{2} - 10 x) + 4 \cdot 9$ .

$4 (x^{2} - 10 x + 9) = 0$

Factor.

Factor $x^{2} - 10 x + 9$ 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Write the factored form using these integers.

$4 ((x - 9) (x - 1)) = 0$

Remove unnecessary parentheses.

$4 (x - 9) (x - 1) = 0$

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

$x - 9 = 0$

$x - 1 = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $4 (x - 9) (x - 1) = 0$ true.

$x = 9, 1$

Solve for x square root of 4x-2 = square root of 2x-2

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