Solve sqrt{3x}+18=x

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

Solve for x

$$x=27$$

Solution Steps

Subtract 18 from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

$$3x=x^{2}-36x+324$$

Subtract x^{2} from both sides.

$$3x-x^{2}=-36x+324$$

Add 36x to both sides.

$$3x-x^{2}+36x=324$$

Combine 3x and 36x to get 39x.

$$39x-x^{2}=324$$

Subtract 324 from both sides.

$$39x-x^{2}-324=0$$

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

$$-x^{2}+39x-324=0$$

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.

$$a+b=39 ab=-\left(-324\right)=324$$

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 39.

$$a=27 b=12$$

Rewrite -x^{2}+39x-324 as \left(-x^{2}+27x\right)+\left(12x-324\right).

$$\left(-x^{2}+27x\right)+\left(12x-324\right)$$

Factor out -x in the first and 12 in the second group.

$$-x\left(x-27\right)+12\left(x-27\right)$$

Factor out common term x-27 by using distributive property.

$$\left(x-27\right)\left(-x+12\right)$$

To find equation solutions, solve x-27=0 and -x+12=0.

$$x=27 x=12$$

Substitute 27 for x in the equation \sqrt{3x}+18=x.

$$\sqrt{3\times 27}+18=27$$

Simplify. The value x=27 satisfies the equation.

$$27=27$$

Substitute 12 for x in the equation \sqrt{3x}+18=x.

$$\sqrt{3\times 12}+18=12$$

Simplify. The value x=12 does not satisfy the equation.

$$24=12$$

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

$$x=27$$