Solve sqrt{9x-20}=x

$$\sqrt{ 9x-20 } =x$$

Solve for x

$$x=5$$

$$x=4$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{9x-20}\right)^{2}=x^{2}$$

Calculate \sqrt{9x-20} to the power of 2 and get 9x-20.

$$9x-20=x^{2}$$

Subtract x^{2} from both sides.

$$9x-20-x^{2}=0$$

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

$$-x^{2}+9x-20=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-20. To find a and b, set up a system to be solved.

$$a+b=9 ab=-\left(-20\right)=20$$

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 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 9.

$$a=5 b=4$$

Rewrite -x^{2}+9x-20 as \left(-x^{2}+5x\right)+\left(4x-20\right).

$$\left(-x^{2}+5x\right)+\left(4x-20\right)$$

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

$$-x\left(x-5\right)+4\left(x-5\right)$$

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

$$\left(x-5\right)\left(-x+4\right)$$

To find equation solutions, solve x-5=0 and -x+4=0.

$$x=5 x=4$$

Substitute 5 for x in the equation \sqrt{9x-20}=x.

$$\sqrt{9\times 5-20}=5$$

Simplify. The value x=5 satisfies the equation.

$$5=5$$

Substitute 4 for x in the equation \sqrt{9x-20}=x.

$$\sqrt{9\times 4-20}=4$$

Simplify. The value x=4 satisfies the equation.

$$4=4$$

List all solutions of \sqrt{9x-20}=x.

$$x=5 x=4$$