Solve x-12/x+12=sqrt{1/2}

$$\frac{ x-12 }{ x+12 } = \sqrt{ \frac{ 1 }{ 2 } }$$

Solve for x

$$x=24\sqrt{2}+36\approx 69.941125497$$

Steps for Solving Linear Equation

Variable x cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by x+12.

$$x-12=\left(x+12\right)\sqrt{\frac{1}{2}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$x-12=\left(x+12\right)\times \frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$x-12=\left(x+12\right)\times \frac{1}{\sqrt{2}}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$x-12=\left(x+12\right)\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$x-12=\left(x+12\right)\times \frac{\sqrt{2}}{2}$$

Express \left(x+12\right)\times \frac{\sqrt{2}}{2} as a single fraction.

$$x-12=\frac{\left(x+12\right)\sqrt{2}}{2}$$

Use the distributive property to multiply x+12 by \sqrt{2}.

$$x-12=\frac{x\sqrt{2}+12\sqrt{2}}{2}$$

Subtract \frac{x\sqrt{2}+12\sqrt{2}}{2} from both sides.

$$x-12-\frac{x\sqrt{2}+12\sqrt{2}}{2}=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$x-\frac{x\sqrt{2}+12\sqrt{2}}{2}=12$$

Multiply both sides of the equation by 2.

$$2x-\left(x\sqrt{2}+12\sqrt{2}\right)=24$$

To find the opposite of x\sqrt{2}+12\sqrt{2}, find the opposite of each term.

$$2x-x\sqrt{2}-12\sqrt{2}=24$$

Add 12\sqrt{2} to both sides.

$$2x-x\sqrt{2}=24+12\sqrt{2}$$

Combine all terms containing x.

$$\left(2-\sqrt{2}\right)x=24+12\sqrt{2}$$

The equation is in standard form.

$$\left(2-\sqrt{2}\right)x=12\sqrt{2}+24$$

Divide both sides by 2-\sqrt{2}.

$$\frac{\left(2-\sqrt{2}\right)x}{2-\sqrt{2}}=\frac{12\sqrt{2}+24}{2-\sqrt{2}}$$

Dividing by 2-\sqrt{2} undoes the multiplication by 2-\sqrt{2}.

$$x=\frac{12\sqrt{2}+24}{2-\sqrt{2}}$$

Divide 24+12\sqrt{2} by 2-\sqrt{2}.

$$x=24\sqrt{2}+36$$