Solve x-3sqrt{x}-18=0

$$x - 3 \sqrt { x } - 18 = 0$$

Solve for x

$$x=36$$

Solution Steps

Add 18 to both sides. Anything plus zero gives itself.

$$x-3\sqrt{x}=18$$

Subtract x from both sides of the equation.

$$-3\sqrt{x}=18-x$$

Square both sides of the equation.

$$\left(-3\sqrt{x}\right)^{2}=\left(18-x\right)^{2}$$

Expand \left(-3\sqrt{x}\right)^{2}.

$$\left(-3\right)^{2}\left(\sqrt{x}\right)^{2}=\left(18-x\right)^{2}$$

Calculate -3 to the power of 2 and get 9.

$$9\left(\sqrt{x}\right)^{2}=\left(18-x\right)^{2}$$

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

$$9x=\left(18-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(18-x\right)^{2}.

$$9x=324-36x+x^{2}$$

Add 36x to both sides.

$$9x+36x=324+x^{2}$$

Combine 9x and 36x to get 45x.

$$45x=324+x^{2}$$

Subtract x^{2} from both sides.

$$45x-x^{2}=324$$

Subtract 324 from both sides.

$$45x-x^{2}-324=0$$

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

$$-x^{2}+45x-324=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-324. To find a and b, set up a system to be solved.

$$a+b=45 ab=-\left(-324\right)=324$$

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 324.

$$1,324 2,162 3,108 4,81 6,54 9,36 12,27 18,18$$

Calculate the sum for each pair.

$$1+324=325 2+162=164 3+108=111 4+81=85 6+54=60 9+36=45 12+27=39 18+18=36$$

The solution is the pair that gives sum 45.

$$a=36 b=9$$

Rewrite -x^{2}+45x-324 as \left(-x^{2}+36x\right)+\left(9x-324\right).

$$\left(-x^{2}+36x\right)+\left(9x-324\right)$$

Factor out -x in the first and 9 in the second group.

$$-x\left(x-36\right)+9\left(x-36\right)$$

Factor out common term x-36 by using distributive property.

$$\left(x-36\right)\left(-x+9\right)$$

To find equation solutions, solve x-36=0 and -x+9=0.

$$x=36 x=9$$

Substitute 36 for x in the equation x-3\sqrt{x}-18=0.

$$36-3\sqrt{36}-18=0$$

Simplify. The value x=36 satisfies the equation.

$$0=0$$

Substitute 9 for x in the equation x-3\sqrt{x}-18=0.

$$9-3\sqrt{9}-18=0$$

Simplify. The value x=9 does not satisfy the equation.

$$-18=0$$

Equation -3\sqrt{x}=18-x has a unique solution.

$$x=36$$