Solve sqrt{13x-23}-1=x

$$\sqrt { 13 x - 23 } - 1 = x$$

Solve for x

$$x=8$$

$$x=3$$

Solution Steps

Subtract -1 from both sides of the equation.

$$\sqrt{13x-23}=x+1$$

Square both sides of the equation.

$$\left(\sqrt{13x-23}\right)^{2}=\left(x+1\right)^{2}$$

Calculate \sqrt{13x-23} to the power of 2 and get 13x-23.

$$13x-23=\left(x+1\right)^{2}$$

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

$$13x-23=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$13x-23-x^{2}=2x+1$$

Subtract 2x from both sides.

$$13x-23-x^{2}-2x=1$$

Combine 13x and -2x to get 11x.

$$11x-23-x^{2}=1$$

Subtract 1 from both sides.

$$11x-23-x^{2}-1=0$$

Subtract 1 from -23 to get -24.

$$11x-24-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+11x-24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=11 ab=-\left(-24\right)=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 11.

$$a=8 b=3$$

Rewrite -x^{2}+11x-24 as \left(-x^{2}+8x\right)+\left(3x-24\right).

$$\left(-x^{2}+8x\right)+\left(3x-24\right)$$

Factor out -x in the first and 3 in the second group.

$$-x\left(x-8\right)+3\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(-x+3\right)$$

To find equation solutions, solve x-8=0 and -x+3=0.

$$x=8 x=3$$

Substitute 8 for x in the equation \sqrt{13x-23}-1=x.

$$\sqrt{13\times 8-23}-1=8$$

Simplify. The value x=8 satisfies the equation.

$$8=8$$

Substitute 3 for x in the equation \sqrt{13x-23}-1=x.

$$\sqrt{13\times 3-23}-1=3$$

Simplify. The value x=3 satisfies the equation.

$$3=3$$

List all solutions of \sqrt{13x-23}=x+1.

$$x=8 x=3$$