Solve (2x-1)^2-12x=2(x-1)(x+1)+6x(x-2)

$$( 2 x - 1 ) ^ { 2 } - 12 x = 2 ( x - 1 ) ( x + 1 ) + 6 x ( x - 2 )$$

Solve for x

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

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

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}.

$$4x^{2}-4x+1-12x=2\left(x-1\right)\left(x+1\right)+6x\left(x-2\right)$$

Combine -4x and -12x to get -16x.

$$4x^{2}-16x+1=2\left(x-1\right)\left(x+1\right)+6x\left(x-2\right)$$

Use the distributive property to multiply 2 by x-1.

$$4x^{2}-16x+1=\left(2x-2\right)\left(x+1\right)+6x\left(x-2\right)$$

Use the distributive property to multiply 2x-2 by x+1 and combine like terms.

$$4x^{2}-16x+1=2x^{2}-2+6x\left(x-2\right)$$

Use the distributive property to multiply 6x by x-2.

$$4x^{2}-16x+1=2x^{2}-2+6x^{2}-12x$$

Combine 2x^{2} and 6x^{2} to get 8x^{2}.

$$4x^{2}-16x+1=8x^{2}-2-12x$$

Subtract 8x^{2} from both sides.

$$4x^{2}-16x+1-8x^{2}=-2-12x$$

Combine 4x^{2} and -8x^{2} to get -4x^{2}.

$$-4x^{2}-16x+1=-2-12x$$

Subtract -2 from both sides.

$$-4x^{2}-16x+1-\left(-2\right)=-12x$$

The opposite of -2 is 2.

$$-4x^{2}-16x+1+2=-12x$$

Add 12x to both sides.

$$-4x^{2}-16x+1+2+12x=0$$

Add 1 and 2 to get 3.

$$-4x^{2}-16x+3+12x=0$$

Combine -16x and 12x to get -4x.

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

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.

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

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -12.

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

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -4.

$$a=2 b=-6$$

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

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

Factor out 2x in the first and 3 in the second group.

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

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

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

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

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

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}.

$$4x^{2}-4x+1-12x=2\left(x-1\right)\left(x+1\right)+6x\left(x-2\right)$$

Combine -4x and -12x to get -16x.

$$4x^{2}-16x+1=2\left(x-1\right)\left(x+1\right)+6x\left(x-2\right)$$

Use the distributive property to multiply 2 by x-1.

$$4x^{2}-16x+1=\left(2x-2\right)\left(x+1\right)+6x\left(x-2\right)$$

Use the distributive property to multiply 2x-2 by x+1 and combine like terms.

$$4x^{2}-16x+1=2x^{2}-2+6x\left(x-2\right)$$

Use the distributive property to multiply 6x by x-2.

$$4x^{2}-16x+1=2x^{2}-2+6x^{2}-12x$$

Combine 2x^{2} and 6x^{2} to get 8x^{2}.

$$4x^{2}-16x+1=8x^{2}-2-12x$$

Subtract 8x^{2} from both sides.

$$4x^{2}-16x+1-8x^{2}=-2-12x$$

Combine 4x^{2} and -8x^{2} to get -4x^{2}.

$$-4x^{2}-16x+1=-2-12x$$

Subtract -2 from both sides.

$$-4x^{2}-16x+1-\left(-2\right)=-12x$$

The opposite of -2 is 2.

$$-4x^{2}-16x+1+2=-12x$$

Add 12x to both sides.

$$-4x^{2}-16x+1+2+12x=0$$

Add 1 and 2 to get 3.

$$-4x^{2}-16x+3+12x=0$$

Combine -16x and 12x to get -4x.

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

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

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

Square -4.

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

Multiply -4 times -4.

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

Multiply 16 times 3.

$$x=\frac{-\left(-4\right)±\sqrt{16+48}}{2\left(-4\right)}$$

Add 16 to 48.

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

Take the square root of 64.

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

The opposite of -4 is 4.

$$x=\frac{4±8}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{4±8}{-8}$$

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

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

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{4±8}{-8} when ± is minus. Subtract 8 from 4.

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

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}.

$$4x^{2}-4x+1-12x=2\left(x-1\right)\left(x+1\right)+6x\left(x-2\right)$$

Combine -4x and -12x to get -16x.

$$4x^{2}-16x+1=2\left(x-1\right)\left(x+1\right)+6x\left(x-2\right)$$

Use the distributive property to multiply 2 by x-1.

$$4x^{2}-16x+1=\left(2x-2\right)\left(x+1\right)+6x\left(x-2\right)$$

Use the distributive property to multiply 2x-2 by x+1 and combine like terms.

$$4x^{2}-16x+1=2x^{2}-2+6x\left(x-2\right)$$

Use the distributive property to multiply 6x by x-2.

$$4x^{2}-16x+1=2x^{2}-2+6x^{2}-12x$$

Combine 2x^{2} and 6x^{2} to get 8x^{2}.

$$4x^{2}-16x+1=8x^{2}-2-12x$$

Subtract 8x^{2} from both sides.

$$4x^{2}-16x+1-8x^{2}=-2-12x$$

Combine 4x^{2} and -8x^{2} to get -4x^{2}.

$$-4x^{2}-16x+1=-2-12x$$

Add 12x to both sides.

$$-4x^{2}-16x+1+12x=-2$$

Combine -16x and 12x to get -4x.

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

Subtract 1 from both sides.

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

Subtract 1 from -2 to get -3.

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

Divide both sides by -4.

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

Dividing by -4 undoes the multiplication by -4.

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

Divide -4 by -4.

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

Divide -3 by -4.

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

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{3}{4}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add \frac{3}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract \frac{1}{2} from both sides of the equation.

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