Solve the Rational Equation for x square root of x-6 = square root of 36-2x

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

Step 1

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

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

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 6}$ as ${(x - 6)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(x - 6)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 6)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${\sqrt{36 - 2 x}}^{2}$ .

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

Factor $2$ out of $36$ .

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x - 6 = 2 (18 - x)$

Apply the distributive property.

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

Multiply.

Multiply $2$ by $18$ .

$x - 6 = 36 + 2 (- x)$

Multiply $- 1$ by $2$ .

$x - 6 = 36 - 2 x$

Step 3

Solve for $x$ .

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

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

$x - 6 + 2 x = 36$

Add $x$ and $2 x$ .

$3 x - 6 = 36$

Move all terms not containing $x$ to the right side of the equation.

Add $6$ to both sides of the equation.

$3 x = 36 + 6$

Add $36$ and $6$ .

$3 x = 42$

Divide each term in $3 x = 42$ by $3$ and simplify.

Divide each term in $3 x = 42$ by $3$ .

$\frac{3 x}{3} = \frac{42}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{42}{3}$

Divide $x$ by $1$ .

$x = \frac{42}{3}$

Simplify the right side.

Divide $42$ by $3$ .

$x = 14$

Solve the Rational Equation for x square root of x-6 = square root of 36-2x

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