Solve sqrt{9x-2}-2x=0

$$\sqrt { 9 x - 2 } - 2 x = 0$$

Solve for x

$$x=2$$

$$x=\frac{1}{4}=0.25$$

Solution Steps

Subtract -2x from both sides of the equation.

$$\sqrt{9x-2}=2x$$

Square both sides of the equation.

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

Calculate \sqrt{9x-2} to the power of 2 and get 9x-2.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$9x-2=4x^{2}$$

Subtract 4x^{2} from both sides.

$$9x-2-4x^{2}=0$$

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

$$-4x^{2}+9x-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=9 ab=-4\left(-2\right)=8$$

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 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 9.

$$a=8 b=1$$

Rewrite -4x^{2}+9x-2 as \left(-4x^{2}+8x\right)+\left(x-2\right).

$$\left(-4x^{2}+8x\right)+\left(x-2\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(-x+2\right)-\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(4x-1\right)$$

To find equation solutions, solve -x+2=0 and 4x-1=0.

$$x=2 x=\frac{1}{4}$$

Substitute 2 for x in the equation \sqrt{9x-2}-2x=0.

$$\sqrt{9\times 2-2}-2\times 2=0$$

Simplify. The value x=2 satisfies the equation.

$$0=0$$

Substitute \frac{1}{4} for x in the equation \sqrt{9x-2}-2x=0.

$$\sqrt{9\times \frac{1}{4}-2}-2\times \frac{1}{4}=0$$

Simplify. The value x=\frac{1}{4} satisfies the equation.

$$0=0$$

List all solutions of \sqrt{9x-2}=2x.

$$x=2 x=\frac{1}{4}$$