Solve sqrt{6-x}-x/sqrt{6-x}=0

$$\sqrt { 6 - x } - \frac { x } { \sqrt { 6 - x } } = 0$$

Solve for x

$$x=3$$

Solution Steps

Subtract -\frac{x}{\sqrt{6-x}} from both sides of the equation.

$$\sqrt{6-x}=-\left(-\frac{x}{\sqrt{6-x}}\right)$$

Multiply -1 and -1 to get 1.

$$\sqrt{6-x}=\frac{x}{\sqrt{6-x}}$$

Square both sides of the equation.

$$\left(\sqrt{6-x}\right)^{2}=\left(\frac{x}{\sqrt{6-x}}\right)^{2}$$

Calculate \sqrt{6-x} to the power of 2 and get 6-x.

$$6-x=\left(\frac{x}{\sqrt{6-x}}\right)^{2}$$

To raise \frac{x}{\sqrt{6-x}} to a power, raise both numerator and denominator to the power and then divide.

$$6-x=\frac{x^{2}}{\left(\sqrt{6-x}\right)^{2}}$$

Calculate \sqrt{6-x} to the power of 2 and get 6-x.

$$6-x=\frac{x^{2}}{6-x}$$

Multiply both sides of the equation by -x+6.

$$\left(-x+6\right)\times 6-x\left(-x+6\right)=x^{2}$$

Use the distributive property to multiply -x+6 by 6.

$$-6x+36-x\left(-x+6\right)=x^{2}$$

Use the distributive property to multiply -x by -x+6.

$$-6x+36+x^{2}-6x=x^{2}$$

Combine -6x and -6x to get -12x.

$$-12x+36+x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$-12x+36+x^{2}-x^{2}=0$$

Combine x^{2} and -x^{2} to get 0.

$$-12x+36=0$$

Subtract 36 from both sides. Anything subtracted from zero gives its negation.

$$-12x=-36$$

Divide both sides by -12.

$$x=\frac{-36}{-12}$$

Divide -36 by -12 to get 3.

$$x=3$$

Substitute 3 for x in the equation \sqrt{6-x}-\frac{x}{\sqrt{6-x}}=0.

$$\sqrt{6-3}-\frac{3}{\sqrt{6-3}}=0$$

Simplify. The value x=3 satisfies the equation.

$$0=0$$

Equation \sqrt{6-x}=\frac{x}{\sqrt{6-x}} has a unique solution.

$$x=3$$