Solve sqrt{3x-11}=x-5

$$\sqrt { 3 x - 11 } = x - 5$$

Solve for x

$$x=9$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{3x-11}\right)^{2}=\left(x-5\right)^{2}$$

Calculate \sqrt{3x-11} to the power of 2 and get 3x-11.

$$3x-11=\left(x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$3x-11=x^{2}-10x+25$$

Subtract x^{2} from both sides.

$$3x-11-x^{2}=-10x+25$$

Add 10x to both sides.

$$3x-11-x^{2}+10x=25$$

Combine 3x and 10x to get 13x.

$$13x-11-x^{2}=25$$

Subtract 25 from both sides.

$$13x-11-x^{2}-25=0$$

Subtract 25 from -11 to get -36.

$$13x-36-x^{2}=0$$

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

$$-x^{2}+13x-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=13 ab=-\left(-36\right)=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 13.

$$a=9 b=4$$

Rewrite -x^{2}+13x-36 as \left(-x^{2}+9x\right)+\left(4x-36\right).

$$\left(-x^{2}+9x\right)+\left(4x-36\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(x-9\right)+4\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(-x+4\right)$$

To find equation solutions, solve x-9=0 and -x+4=0.

$$x=9 x=4$$

Substitute 9 for x in the equation \sqrt{3x-11}=x-5.

$$\sqrt{3\times 9-11}=9-5$$

Simplify. The value x=9 satisfies the equation.

$$4=4$$

Substitute 4 for x in the equation \sqrt{3x-11}=x-5.

$$\sqrt{3\times 4-11}=4-5$$

Simplify. The value x=4 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Equation \sqrt{3x-11}=x-5 has a unique solution.

$$x=9$$