Solve sqrt{3x+2}=3x

$$\sqrt { 3 x + 2 } = 3 x$$

Solve for x

$$x=\frac{2}{3}\approx 0.666666667$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$3x+2=9x^{2}$$

Subtract 9x^{2} from both sides.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=3 ab=-9\times 2=-18$$

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 -18.

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

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 3.

$$a=6 b=-3$$

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

$$\left(-9x^{2}+6x\right)+\left(-3x+2\right)$$

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

$$-3x\left(3x-2\right)-\left(3x-2\right)$$

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(-3x-1\right)$$

To find equation solutions, solve 3x-2=0 and -3x-1=0.

$$x=\frac{2}{3} x=-\frac{1}{3}$$

Substitute \frac{2}{3} for x in the equation \sqrt{3x+2}=3x.

$$\sqrt{3\times \frac{2}{3}+2}=3\times \frac{2}{3}$$

Simplify. The value x=\frac{2}{3} satisfies the equation.

$$2=2$$

Substitute -\frac{1}{3} for x in the equation \sqrt{3x+2}=3x.

$$\sqrt{3\left(-\frac{1}{3}\right)+2}=3\left(-\frac{1}{3}\right)$$

Simplify. The value x=-\frac{1}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Equation \sqrt{3x+2}=3x has a unique solution.

$$x=\frac{2}{3}$$