Solve by Factoring x-2 square root of 2x=16

$x - 2 \sqrt{2 x} = 16$

Step 1

Subtract $16$ from both sides of the equation.

$x - 2 \sqrt{2 x} - 16 = 0$

Step 2

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

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{2 x} - 16 = - x$

Add $16$ to both sides of the equation.

$- 2 \sqrt{2 x} = - x + 16$

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $2 x$ .

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

Multiply $- 2 \cdot 2^{\frac{1}{2}}$ .

Factor out negative.

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

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

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

${(- 2^{\frac{3}{2}} x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

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

Apply the product rule to $- 2^{\frac{3}{2}} x^{\frac{1}{2}}$ .

${(- 2^{\frac{3}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

Apply the product rule to $- 2^{\frac{3}{2}}$ .

${(- 1)}^{2} {(2^{\frac{3}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

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

$1 {(2^{\frac{3}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

Multiply ${(2^{\frac{3}{2}})}^{2}$ by $1$ .

${(2^{\frac{3}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

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

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

$2^{\frac{3}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2^{\frac{3}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

Rewrite the expression.

$2^{3} {(x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

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

$8 {(x^{\frac{1}{2}})}^{2} = {(- x + 16)}^{2}$

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

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

$8 x^{\frac{1}{2} \cdot 2} = {(- x + 16)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$8 x^{\frac{1}{2} \cdot 2} = {(- x + 16)}^{2}$

Rewrite the expression.

$8 x^{1} = {(- x + 16)}^{2}$

Simplify.

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

Simplify the right side.

Simplify ${(- x + 16)}^{2}$ .

Rewrite ${(- x + 16)}^{2}$ as $(- x + 16) (- x + 16)$ .

$8 x = (- x + 16) (- x + 16)$

Expand $(- x + 16) (- x + 16)$ using the FOIL Method.

Apply the distributive property.

$8 x = - x (- x + 16) + 16 (- x + 16)$

Apply the distributive property.

$8 x = - x (- x) - x \cdot 16 + 16 (- x + 16)$

Apply the distributive property.

$8 x = - x (- x) - x \cdot 16 + 16 (- x) + 16 \cdot 16$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$8 x = - 1 \cdot - 1 x \cdot x - x \cdot 16 + 16 (- x) + 16 \cdot 16$

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

Move $x$ .

$8 x = - 1 \cdot - 1 (x \cdot x) - x \cdot 16 + 16 (- x) + 16 \cdot 16$

Multiply $x$ by $x$ .

$8 x = - 1 \cdot - 1 x^{2} - x \cdot 16 + 16 (- x) + 16 \cdot 16$

Multiply $- 1$ by $- 1$ .

$8 x = 1 x^{2} - x \cdot 16 + 16 (- x) + 16 \cdot 16$

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

$8 x = x^{2} - x \cdot 16 + 16 (- x) + 16 \cdot 16$

Multiply $16$ by $- 1$ .

$8 x = x^{2} - 16 x + 16 (- x) + 16 \cdot 16$

Multiply $- 1$ by $16$ .

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

Multiply $16$ by $16$ .

$8 x = x^{2} - 16 x - 16 x + 256$

Subtract $16 x$ from $- 16 x$ .

$8 x = x^{2} - 32 x + 256$

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.

$x^{2} - 32 x + 256 = 8 x$

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

Subtract $8 x$ from both sides of the equation.

$x^{2} - 32 x + 256 - 8 x = 0$

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

$x^{2} - 40 x + 256 = 0$

Factor $x^{2} - 40 x + 256$ 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 $256$ and whose sum is $- 40$ .

$- 32, - 8$

Write the factored form using these integers.

$(x - 32) (x - 8) = 0$

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

$x - 32 = 0$

$x - 8 = 0$

Set $x - 32$ equal to $0$ and solve for $x$ .

Set $x - 32$ equal to $0$ .

$x - 32 = 0$

Add $32$ to both sides of the equation.

$x = 32$

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.

$x = 8$

The final solution is all the values that make $(x - 32) (x - 8) = 0$ true.

$x = 32, 8$

Step 6

Exclude the solutions that do not make $x - 2 \sqrt{2 x} - 16 = 0$ true.

$x = 32$

Solve by Factoring x-2 square root of 2x=16

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