Solve (6x^2-28x+16)

$$( 6 x ^ { 2 } - 28 x + 16 )$$

Factor

$$2\left(x-4\right)\left(3x-2\right)$$

Solution Steps

Factor out 2.

$$2\left(3x^{2}-14x+8\right)$$

Consider 3x^{2}-14x+8. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-14 ab=3\times 8=24$$

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 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -14.

$$a=-12 b=-2$$

Rewrite 3x^{2}-14x+8 as \left(3x^{2}-12x\right)+\left(-2x+8\right).

$$\left(3x^{2}-12x\right)+\left(-2x+8\right)$$

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

$$3x\left(x-4\right)-2\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(3x-2\right)$$

Rewrite the complete factored expression.

$$2\left(x-4\right)\left(3x-2\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$6x^{2}-28x+16=0$$

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.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 6\times 16}}{2\times 6}$$

Square -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 6\times 16}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-28\right)±\sqrt{784-24\times 16}}{2\times 6}$$

Multiply -24 times 16.

$$x=\frac{-\left(-28\right)±\sqrt{784-384}}{2\times 6}$$

Add 784 to -384.

$$x=\frac{-\left(-28\right)±\sqrt{400}}{2\times 6}$$

Take the square root of 400.

$$x=\frac{-\left(-28\right)±20}{2\times 6}$$

The opposite of -28 is 28.

$$x=\frac{28±20}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{28±20}{12}$$

Now solve the equation x=\frac{28±20}{12} when ± is plus. Add 28 to 20.

$$x=\frac{48}{12}$$

Divide 48 by 12.

$$x=4$$

Now solve the equation x=\frac{28±20}{12} when ± is minus. Subtract 20 from 28.

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

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

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and \frac{2}{3} for x_{2}.

$$6x^{2}-28x+16=6\left(x-4\right)\left(x-\frac{2}{3}\right)$$

Subtract \frac{2}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$6x^{2}-28x+16=6\left(x-4\right)\times \frac{3x-2}{3}$$

Cancel out 3, the greatest common factor in 6 and 3.

$$6x^{2}-28x+16=2\left(x-4\right)\left(3x-2\right)$$

Evaluate

$$2\left(x-4\right)\left(3x-2\right)$$