Solve 6x-sqrt{18x-8}=2

$$6 x - \sqrt { 18 x - 8 } = 2$$

Solve for x

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

$$x=\frac{1}{2}=0.5$$

Solution Steps

Subtract 6x from both sides of the equation.

$$-\sqrt{18x-8}=2-6x$$

Square both sides of the equation.

$$\left(-\sqrt{18x-8}\right)^{2}=\left(2-6x\right)^{2}$$

Expand \left(-\sqrt{18x-8}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{18x-8}\right)^{2}=\left(2-6x\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{18x-8}\right)^{2}=\left(2-6x\right)^{2}$$

Calculate \sqrt{18x-8} to the power of 2 and get 18x-8.

$$1\left(18x-8\right)=\left(2-6x\right)^{2}$$

Use the distributive property to multiply 1 by 18x-8.

$$18x-8=\left(2-6x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-6x\right)^{2}.

$$18x-8=4-24x+36x^{2}$$

Subtract 4 from both sides.

$$18x-8-4=-24x+36x^{2}$$

Subtract 4 from -8 to get -12.

$$18x-12=-24x+36x^{2}$$

Add 24x to both sides.

$$18x-12+24x=36x^{2}$$

Combine 18x and 24x to get 42x.

$$42x-12=36x^{2}$$

Subtract 36x^{2} from both sides.

$$42x-12-36x^{2}=0$$

Divide both sides by 6.

$$7x-2-6x^{2}=0$$

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

$$-6x^{2}+7x-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -6x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=7 ab=-6\left(-2\right)=12$$

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 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 7.

$$a=4 b=3$$

Rewrite -6x^{2}+7x-2 as \left(-6x^{2}+4x\right)+\left(3x-2\right).

$$\left(-6x^{2}+4x\right)+\left(3x-2\right)$$

Factor out 2x in the first and -1 in the second group.

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

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

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

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

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

Substitute \frac{2}{3} for x in the equation 6x-\sqrt{18x-8}=2.

$$6\times \frac{2}{3}-\sqrt{18\times \frac{2}{3}-8}=2$$

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

$$2=2$$

Substitute \frac{1}{2} for x in the equation 6x-\sqrt{18x-8}=2.

$$6\times \frac{1}{2}-\sqrt{18\times \frac{1}{2}-8}=2$$

Simplify. The value x=\frac{1}{2} satisfies the equation.

$$2=2$$

List all solutions of -\sqrt{18x-8}=2-6x.

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