Solve 2x=sqrt{9x-2}

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

Solve for x

$$x=2$$

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

Solution Steps

Square both sides of the equation.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$4x^{2}=\left(\sqrt{9x-2}\right)^{2}$$

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

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

Subtract 9x from both sides.

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

Add 2 to both sides.

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

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.

$$a+b=-9 ab=4\times 2=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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 2x=\sqrt{9x-2}.

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

Simplify. The value x=2 satisfies the equation.

$$4=4$$

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

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

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

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

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

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