Solve sqrt{3x+54}-x=0

$$\sqrt { 3 x + 54 } - x = 0$$

Solve for x

$$x=9$$

Solution Steps

Subtract -x from both sides of the equation.

$$\sqrt{3x+54}=x$$

Square both sides of the equation.

$$\left(\sqrt{3x+54}\right)^{2}=x^{2}$$

Calculate \sqrt{3x+54} to the power of 2 and get 3x+54.

$$3x+54=x^{2}$$

Subtract x^{2} from both sides.

$$3x+54-x^{2}=0$$

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

$$-x^{2}+3x+54=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+54. To find a and b, set up a system to be solved.

$$a+b=3 ab=-54=-54$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -54.

$$-1,54 -2,27 -3,18 -6,9$$

Calculate the sum for each pair.

$$-1+54=53 -2+27=25 -3+18=15 -6+9=3$$

The solution is the pair that gives sum 3.

$$a=9 b=-6$$

Rewrite -x^{2}+3x+54 as \left(-x^{2}+9x\right)+\left(-6x+54\right).

$$\left(-x^{2}+9x\right)+\left(-6x+54\right)$$

Factor out -x in the first and -6 in the second group.

$$-x\left(x-9\right)-6\left(x-9\right)$$

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

$$\left(x-9\right)\left(-x-6\right)$$

To find equation solutions, solve x-9=0 and -x-6=0.

$$x=9 x=-6$$

Substitute 9 for x in the equation \sqrt{3x+54}-x=0.

$$\sqrt{3\times 9+54}-9=0$$

Simplify. The value x=9 satisfies the equation.

$$0=0$$

Substitute -6 for x in the equation \sqrt{3x+54}-x=0.

$$\sqrt{3\left(-6\right)+54}-\left(-6\right)=0$$

Simplify. The value x=-6 does not satisfy the equation.

$$12=0$$

Equation \sqrt{3x+54}=x has a unique solution.

$$x=9$$