Find All Complex Solutions square root of 2x- square root of 36-2x=6

$\sqrt{2 x} - \sqrt{36 - 2 x} = 6$

Solve for $\sqrt{2 x}$ .

Factor $2$ out of $36 - 2 x$ .

Factor $2$ out of $36$ .

$\sqrt{2 x} - \sqrt{2 (18) - 2 x} = 6$

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

$\sqrt{2 x} - \sqrt{2 (18) + 2 (- x)} = 6$

Factor $2$ out of $2 (18) + 2 (- x)$ .

$\sqrt{2 x} - \sqrt{2 (18 - x)} = 6$

Add $\sqrt{2 (18 - x)}$ to both sides of the equation.

$\sqrt{2 x} = 6 + \sqrt{2 (18 - x)}$

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

${\sqrt{2 x}}^{2} = {(6 + \sqrt{2 (18 - x)})}^{2}$

Simplify each side of the equation.

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

${({(2 x)}^{\frac{1}{2}})}^{2} = {(6 + \sqrt{2 (18 - x)})}^{2}$

Simplify the left side.

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

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

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

${(2 x)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{2 (18 - x)})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{2 (18 - x)})}^{2}$

Rewrite the expression.

${(2 x)}^{1} = {(6 + \sqrt{2 (18 - x)})}^{2}$

Simplify.

$2 x = {(6 + \sqrt{2 (18 - x)})}^{2}$

Simplify the right side.

Simplify ${(6 + \sqrt{2 (18 - x)})}^{2}$ .

Rewrite ${(6 + \sqrt{2 (18 - x)})}^{2}$ as $(6 + \sqrt{2 (18 - x)}) (6 + \sqrt{2 (18 - x)})$ .

$2 x = (6 + \sqrt{2 (18 - x)}) (6 + \sqrt{2 (18 - x)})$

Expand $(6 + \sqrt{2 (18 - x)}) (6 + \sqrt{2 (18 - x)})$ using the FOIL Method.

Apply the distributive property.

$2 x = 6 (6 + \sqrt{2 (18 - x)}) + \sqrt{2 (18 - x)} (6 + \sqrt{2 (18 - x)})$

Apply the distributive property.

$2 x = 6 \cdot 6 + 6 \sqrt{2 (18 - x)} + \sqrt{2 (18 - x)} (6 + \sqrt{2 (18 - x)})$

Apply the distributive property.

$2 x = 6 \cdot 6 + 6 \sqrt{2 (18 - x)} + \sqrt{2 (18 - x)} \cdot 6 + \sqrt{2 (18 - x)} \sqrt{2 (18 - x)}$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + \sqrt{2 (18 - x)} \cdot 6 + \sqrt{2 (18 - x)} \sqrt{2 (18 - x)}$

Move $6$ to the left of $\sqrt{2 (18 - x)}$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \cdot \sqrt{2 (18 - x)} + \sqrt{2 (18 - x)} \sqrt{2 (18 - x)}$

Multiply $\sqrt{2 (18 - x)} \sqrt{2 (18 - x)}$ .

Raise $\sqrt{2 (18 - x)}$ to the power of $1$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {\sqrt{2 (18 - x)}}^{1} \sqrt{2 (18 - x)}$

Raise $\sqrt{2 (18 - x)}$ to the power of $1$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {\sqrt{2 (18 - x)}}^{1} {\sqrt{2 (18 - x)}}^{1}$

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

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {\sqrt{2 (18 - x)}}^{1 + 1}$

Add $1$ and $1$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {\sqrt{2 (18 - x)}}^{2}$

Rewrite ${\sqrt{2 (18 - x)}}^{2}$ as $2 (18 - x)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (18 - x)}$ as ${(2 (18 - x))}^{\frac{1}{2}}$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {({(2 (18 - x))}^{\frac{1}{2}})}^{2}$

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

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {(2 (18 - x))}^{\frac{1}{2} \cdot 2}$

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

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {(2 (18 - x))}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {(2 (18 - x))}^{\frac{2}{2}}$

Rewrite the expression.

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + {(2 (18 - x))}^{1}$

Simplify.

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + 2 (18 - x)$

Apply the distributive property.

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + 2 \cdot 18 + 2 (- x)$

Multiply $2$ by $18$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + 36 + 2 (- x)$

Multiply $- 1$ by $2$ .

$2 x = 36 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} + 36 - 2 x$

Add $36$ and $36$ .

$2 x = 72 + 6 \sqrt{2 (18 - x)} + 6 \sqrt{2 (18 - x)} - 2 x$

Add $6 \sqrt{2 (18 - x)}$ and $6 \sqrt{2 (18 - x)}$ .

$2 x = 72 + 12 \sqrt{2 (18 - x)} - 2 x$

Solve for $12 \sqrt{2 (18 - x)}$ .

Rewrite the equation as $72 + 12 \sqrt{2 (18 - x)} - 2 x = 2 x$ .

$72 + 12 \sqrt{2 (18 - x)} - 2 x = 2 x$

Move all terms not containing $12 \sqrt{2 (18 - x)}$ to the right side of the equation.

Subtract $72$ from both sides of the equation.

$12 \sqrt{2 (18 - x)} - 2 x = 2 x - 72$

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

$12 \sqrt{2 (18 - x)} = 2 x - 72 + 2 x$

Add $2 x$ and $2 x$ .

$12 \sqrt{2 (18 - x)} = 4 x - 72$

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

${(12 \sqrt{2 (18 - x)})}^{2} = {(4 x - 72)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (18 - x)}$ as ${(2 (18 - x))}^{\frac{1}{2}}$ .

${(12 {(2 (18 - x))}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Simplify the left side.

Simplify ${(12 {(2 (18 - x))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(12 {(2 \cdot 18 + 2 (- x))}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Simplify the expression.

Multiply $2$ by $18$ .

${(12 {(36 + 2 (- x))}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Multiply $- 1$ by $2$ .

${(12 {(36 - 2 x)}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Apply the product rule to $12 {(36 - 2 x)}^{\frac{1}{2}}$ .

$12^{2} {({(36 - 2 x)}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Raise $12$ to the power of $2$ .

$144 {({(36 - 2 x)}^{\frac{1}{2}})}^{2} = {(4 x - 72)}^{2}$

Multiply the exponents in ${({(36 - 2 x)}^{\frac{1}{2}})}^{2}$ .

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

$144 {(36 - 2 x)}^{\frac{1}{2} \cdot 2} = {(4 x - 72)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$144 {(36 - 2 x)}^{\frac{1}{2} \cdot 2} = {(4 x - 72)}^{2}$

Rewrite the expression.

$144 {(36 - 2 x)}^{1} = {(4 x - 72)}^{2}$

Simplify.

$144 (36 - 2 x) = {(4 x - 72)}^{2}$

Apply the distributive property.

$144 \cdot 36 + 144 (- 2 x) = {(4 x - 72)}^{2}$

Multiply.

Multiply $144$ by $36$ .

$5184 + 144 (- 2 x) = {(4 x - 72)}^{2}$

Multiply $- 2$ by $144$ .

$5184 - 288 x = {(4 x - 72)}^{2}$

Simplify the right side.

Simplify ${(4 x - 72)}^{2}$ .

Rewrite ${(4 x - 72)}^{2}$ as $(4 x - 72) (4 x - 72)$ .

$5184 - 288 x = (4 x - 72) (4 x - 72)$

Expand $(4 x - 72) (4 x - 72)$ using the FOIL Method.

Apply the distributive property.

$5184 - 288 x = 4 x (4 x - 72) - 72 (4 x - 72)$

Apply the distributive property.

$5184 - 288 x = 4 x (4 x) + 4 x \cdot - 72 - 72 (4 x - 72)$

Apply the distributive property.

$5184 - 288 x = 4 x (4 x) + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$5184 - 288 x = 4 \cdot 4 x \cdot x + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

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

Move $x$ .

$5184 - 288 x = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Multiply $x$ by $x$ .

$5184 - 288 x = 4 \cdot 4 x^{2} + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Multiply $4$ by $4$ .

$5184 - 288 x = 16 x^{2} + 4 x \cdot - 72 - 72 (4 x) - 72 \cdot - 72$

Multiply $- 72$ by $4$ .

$5184 - 288 x = 16 x^{2} - 288 x - 72 (4 x) - 72 \cdot - 72$

Multiply $4$ by $- 72$ .

$5184 - 288 x = 16 x^{2} - 288 x - 288 x - 72 \cdot - 72$

Multiply $- 72$ by $- 72$ .

$5184 - 288 x = 16 x^{2} - 288 x - 288 x + 5184$

Subtract $288 x$ from $- 288 x$ .

$5184 - 288 x = 16 x^{2} - 576 x + 5184$

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.

$16 x^{2} - 576 x + 5184 = 5184 - 288 x$

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

Add $288 x$ to both sides of the equation.

$16 x^{2} - 576 x + 5184 + 288 x = 5184$

Add $- 576 x$ and $288 x$ .

$16 x^{2} - 288 x + 5184 = 5184$

Subtract $5184$ from both sides of the equation.

$16 x^{2} - 288 x + 5184 - 5184 = 0$

Combine the opposite terms in $16 x^{2} - 288 x + 5184 - 5184$ .

Subtract $5184$ from $5184$ .

$16 x^{2} - 288 x + 0 = 0$

Add $16 x^{2} - 288 x$ and $0$ .

$16 x^{2} - 288 x = 0$

Factor $16 x$ out of $16 x^{2} - 288 x$ .

Factor $16 x$ out of $16 x^{2}$ .

$16 x (x) - 288 x = 0$

Factor $16 x$ out of $- 288 x$ .

$16 x (x) + 16 x (- 18) = 0$

Factor $16 x$ out of $16 x (x) + 16 x (- 18)$ .

$16 x (x - 18) = 0$

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

$x = 0$

$x - 18 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 18$ equal to $0$ and solve for $x$ .

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Add $18$ to both sides of the equation.

$x = 18$

The final solution is all the values that make $16 x (x - 18) = 0$ true.

$x = 0, 18$

Exclude the solutions that do not make $\sqrt{2 x} - \sqrt{36 - 2 x} = 6$ true.

$x = 18$

Find All Complex Solutions square root of 2x- square root of 36-2x=6

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