Solve left(2x-1right)^2=4x-2

$${ \left(2x-1 \right) }^{ 2 } = 4x-2$$

Solve for x

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

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1=4x-2$$

Subtract 4x from both sides.

$$4x^{2}-4x+1-4x=-2$$

Combine -4x and -4x to get -8x.

$$4x^{2}-8x+1=-2$$

Add 2 to both sides.

$$4x^{2}-8x+1+2=0$$

Add 1 and 2 to get 3.

$$4x^{2}-8x+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=-8 ab=4\times 3=12$$

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 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 -8.

$$a=-6 b=-2$$

Rewrite 4x^{2}-8x+3 as \left(4x^{2}-6x\right)+\left(-2x+3\right).

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

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

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

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

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

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

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

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1=4x-2$$

Subtract 4x from both sides.

$$4x^{2}-4x+1-4x=-2$$

Combine -4x and -4x to get -8x.

$$4x^{2}-8x+1=-2$$

Add 2 to both sides.

$$4x^{2}-8x+1+2=0$$

Add 1 and 2 to get 3.

$$4x^{2}-8x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -8 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 4\times 3}}{2\times 4}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 4\times 3}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-8\right)±\sqrt{64-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$x=\frac{-\left(-8\right)±\sqrt{64-48}}{2\times 4}$$

Add 64 to -48.

$$x=\frac{-\left(-8\right)±\sqrt{16}}{2\times 4}$$

Take the square root of 16.

$$x=\frac{-\left(-8\right)±4}{2\times 4}$$

The opposite of -8 is 8.

$$x=\frac{8±4}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{8±4}{8}$$

Now solve the equation x=\frac{8±4}{8} when ± is plus. Add 8 to 4.

$$x=\frac{12}{8}$$

Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{8±4}{8} when ± is minus. Subtract 4 from 8.

$$x=\frac{4}{8}$$

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

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

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1=4x-2$$

Subtract 4x from both sides.

$$4x^{2}-4x+1-4x=-2$$

Combine -4x and -4x to get -8x.

$$4x^{2}-8x+1=-2$$

Subtract 1 from both sides.

$$4x^{2}-8x=-2-1$$

Subtract 1 from -2 to get -3.

$$4x^{2}-8x=-3$$

Divide both sides by 4.

$$\frac{4x^{2}-8x}{4}=-\frac{3}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{8}{4}\right)x=-\frac{3}{4}$$

Divide -8 by 4.

$$x^{2}-2x=-\frac{3}{4}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-\frac{3}{4}+1$$

Add -\frac{3}{4} to 1.

$$x^{2}-2x+1=\frac{1}{4}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

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

Add 1 to both sides of the equation.

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