Solve 16x^2-32x+12=0

$$16 { x }^{ 2 } -32x+12=0$$

Solve for x

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

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

Steps Using Factoring By Grouping

Divide both sides by 4.

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

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$16x^{2}-32x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -32 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 16\times 12}}{2\times 16}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 16\times 12}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-32\right)±\sqrt{1024-64\times 12}}{2\times 16}$$

Multiply -64 times 12.

$$x=\frac{-\left(-32\right)±\sqrt{1024-768}}{2\times 16}$$

Add 1024 to -768.

$$x=\frac{-\left(-32\right)±\sqrt{256}}{2\times 16}$$

Take the square root of 256.

$$x=\frac{-\left(-32\right)±16}{2\times 16}$$

The opposite of -32 is 32.

$$x=\frac{32±16}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{32±16}{32}$$

Now solve the equation x=\frac{32±16}{32} when ± is plus. Add 32 to 16.

$$x=\frac{48}{32}$$

Reduce the fraction \frac{48}{32} to lowest terms by extracting and canceling out 16.

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

Now solve the equation x=\frac{32±16}{32} when ± is minus. Subtract 16 from 32.

$$x=\frac{16}{32}$$

Reduce the fraction \frac{16}{32} to lowest terms by extracting and canceling out 16.

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

The equation is now solved.

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

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$16x^{2}-32x+12=0$$

Subtract 12 from both sides of the equation.

$$16x^{2}-32x+12-12=-12$$

Subtracting 12 from itself leaves 0.

$$16x^{2}-32x=-12$$

Divide both sides by 16.

$$\frac{16x^{2}-32x}{16}=-\frac{12}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{32}{16}\right)x=-\frac{12}{16}$$

Divide -32 by 16.

$$x^{2}-2x=-\frac{12}{16}$$

Reduce the fraction \frac{-12}{16} to lowest terms by extracting and canceling out 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.

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