Solve sqrt{3-4x}=6x-2

$$\sqrt { 3 - 4 x } = 6 x - 2$$

Solve for x

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

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{3-4x} to the power of 2 and get 3-4x.

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

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

$$3-4x=36x^{2}-24x+4$$

Subtract 36x^{2} from both sides.

$$3-4x-36x^{2}=-24x+4$$

Add 24x to both sides.

$$3-4x-36x^{2}+24x=4$$

Combine -4x and 24x to get 20x.

$$3+20x-36x^{2}=4$$

Subtract 4 from both sides.

$$3+20x-36x^{2}-4=0$$

Subtract 4 from 3 to get -1.

$$-1+20x-36x^{2}=0$$

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

$$-36x^{2}+20x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -36x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=20 ab=-36\left(-1\right)=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 20.

$$a=18 b=2$$

Rewrite -36x^{2}+20x-1 as \left(-36x^{2}+18x\right)+\left(2x-1\right).

$$\left(-36x^{2}+18x\right)+\left(2x-1\right)$$

Factor out -18x in -36x^{2}+18x.

$$-18x\left(2x-1\right)+2x-1$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(-18x+1\right)$$

To find equation solutions, solve 2x-1=0 and -18x+1=0.

$$x=\frac{1}{2} x=\frac{1}{18}$$

Substitute \frac{1}{2} for x in the equation \sqrt{3-4x}=6x-2.

$$\sqrt{3-4\times \frac{1}{2}}=6\times \frac{1}{2}-2$$

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

$$1=1$$

Substitute \frac{1}{18} for x in the equation \sqrt{3-4x}=6x-2.

$$\sqrt{3-4\times \frac{1}{18}}=6\times \frac{1}{18}-2$$

Simplify. The value x=\frac{1}{18} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{5}{3}=-\frac{5}{3}$$

Equation \sqrt{3-4x}=6x-2 has a unique solution.

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