Solve sqrt{12x-3}=3x

$$\sqrt { 12 x - 3 } = 3 x$$

Solve for x

$$x=1$$

$$x=\frac{1}{3}\approx 0.333333333$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{12x-3}\right)^{2}=\left(3x\right)^{2}$$

Calculate \sqrt{12x-3} to the power of 2 and get 12x-3.

$$12x-3=\left(3x\right)^{2}$$

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

$$12x-3=3^{2}x^{2}$$

Calculate 3 to the power of 2 and get 9.

$$12x-3=9x^{2}$$

Subtract 9x^{2} from both sides.

$$12x-3-9x^{2}=0$$

Divide both sides by 3.

$$4x-1-3x^{2}=0$$

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

$$-3x^{2}+4x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=4 ab=-3\left(-1\right)=3$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=3 b=1$$

Rewrite -3x^{2}+4x-1 as \left(-3x^{2}+3x\right)+\left(x-1\right).

$$\left(-3x^{2}+3x\right)+\left(x-1\right)$$

Factor out 3x in the first and -1 in the second group.

$$3x\left(-x+1\right)-\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(3x-1\right)$$

To find equation solutions, solve -x+1=0 and 3x-1=0.

$$x=1 x=\frac{1}{3}$$

Substitute 1 for x in the equation \sqrt{12x-3}=3x.

$$\sqrt{12\times 1-3}=3\times 1$$

Simplify. The value x=1 satisfies the equation.

$$3=3$$

Substitute \frac{1}{3} for x in the equation \sqrt{12x-3}=3x.

$$\sqrt{12\times \frac{1}{3}-3}=3\times \frac{1}{3}$$

Simplify. The value x=\frac{1}{3} satisfies the equation.

$$1=1$$

List all solutions of \sqrt{12x-3}=3x.

$$x=1 x=\frac{1}{3}$$