Solve by Factoring x- square root of 2x=4

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

Step 1

Subtract $4$ from both sides of the equation.

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

Step 2

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

Subtract $x$ from both sides of the equation.

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

Add $4$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $2 x$ .

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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

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

Apply the product rule to $- 2^{\frac{1}{2}}$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply ${(2^{\frac{1}{2}})}^{2}$ by $1$ .

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

$2 x = - x (- x) - x \cdot 4 + 4 (- x + 4)$

Apply the distributive property.

$2 x = - x (- x) - x \cdot 4 + 4 (- x) + 4 \cdot 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x^{2}$ by $1$ .

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

Multiply $4$ by $- 1$ .

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

Multiply $- 1$ by $4$ .

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

Multiply $4$ by $4$ .

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

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

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

Step 5

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.

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

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

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

$x^{2} - 8 x + 16 - 2 x = 0$

Subtract $2 x$ from $- 8 x$ .

$x^{2} - 10 x + 16 = 0$

Factor $x^{2} - 10 x + 16$ 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 $16$ and whose sum is $- 10$ .

$- 8, - 2$

Write the factored form using these integers.

$(x - 8) (x - 2) = 0$

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

$x - 8 = 0$

$x - 2 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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 $(x - 8) (x - 2) = 0$ true.

$x = 8, 2$

Step 6

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

$x = 8$

Solve by Factoring x- square root of 2x=4

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