Solve sqrt{x-1}=x-1/sqrt{9-x}

$$\sqrt { x - 1 } = \frac { x - 1 } { \sqrt { 9 - x } }$$

Solve for x

$$x=5$$

$$x=1$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x-1}\right)^{2}=\left(\frac{x-1}{\sqrt{9-x}}\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$x-1=\left(\frac{x-1}{\sqrt{9-x}}\right)^{2}$$

To raise \frac{x-1}{\sqrt{9-x}} to a power, raise both numerator and denominator to the power and then divide.

$$x-1=\frac{\left(x-1\right)^{2}}{\left(\sqrt{9-x}\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x-1=\frac{x^{2}-2x+1}{\left(\sqrt{9-x}\right)^{2}}$$

Calculate \sqrt{9-x} to the power of 2 and get 9-x.

$$x-1=\frac{x^{2}-2x+1}{9-x}$$

Multiply both sides of the equation by -x+9.

$$\left(-x+9\right)x+\left(-x+9\right)\left(-1\right)=x^{2}-2x+1$$

Use the distributive property to multiply -x+9 by x.

$$-x^{2}+9x+\left(-x+9\right)\left(-1\right)=x^{2}-2x+1$$

Use the distributive property to multiply -x+9 by -1.

$$-x^{2}+9x+x-9=x^{2}-2x+1$$

Combine 9x and x to get 10x.

$$-x^{2}+10x-9=x^{2}-2x+1$$

Subtract x^{2} from both sides.

$$-x^{2}+10x-9-x^{2}=-2x+1$$

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

$$-2x^{2}+10x-9=-2x+1$$

Add 2x to both sides.

$$-2x^{2}+10x-9+2x=1$$

Combine 10x and 2x to get 12x.

$$-2x^{2}+12x-9=1$$

Subtract 1 from both sides.

$$-2x^{2}+12x-9-1=0$$

Subtract 1 from -9 to get -10.

$$-2x^{2}+12x-10=0$$

Divide both sides by 2.

$$-x^{2}+6x-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=6 ab=-\left(-5\right)=5$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=5 b=1$$

Rewrite -x^{2}+6x-5 as \left(-x^{2}+5x\right)+\left(x-5\right).

$$\left(-x^{2}+5x\right)+\left(x-5\right)$$

Factor out -x in -x^{2}+5x.

$$-x\left(x-5\right)+x-5$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x+1\right)$$

To find equation solutions, solve x-5=0 and -x+1=0.

$$x=5 x=1$$

Substitute 5 for x in the equation \sqrt{x-1}=\frac{x-1}{\sqrt{9-x}}.

$$\sqrt{5-1}=\frac{5-1}{\sqrt{9-5}}$$

Simplify. The value x=5 satisfies the equation.

$$2=2$$

Substitute 1 for x in the equation \sqrt{x-1}=\frac{x-1}{\sqrt{9-x}}.

$$\sqrt{1-1}=\frac{1-1}{\sqrt{9-1}}$$

Simplify. The value x=1 satisfies the equation.

$$0=0$$

List all solutions of \sqrt{x-1}=\frac{x-1}{\sqrt{9-x}}.

$$x=5 x=1$$