Solve 3/4x^2-16x+64

$$\frac { 3 } { 4 } x ^ { 2 } - 16 x + 64$$

Factor

$$\frac{\left(x-16\right)\left(3x-16\right)}{4}$$

Solution Steps

Factor out \frac{1}{4}.

$$\frac{3x^{2}-64x+256}{4}$$

Consider 3x^{2}-64x+256. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+256. To find a and b, set up a system to be solved.

$$a+b=-64 ab=3\times 256=768$$

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 768.

$$-1,-768 -2,-384 -3,-256 -4,-192 -6,-128 -8,-96 -12,-64 -16,-48 -24,-32$$

Calculate the sum for each pair.

$$-1-768=-769 -2-384=-386 -3-256=-259 -4-192=-196 -6-128=-134 -8-96=-104 -12-64=-76 -16-48=-64 -24-32=-56$$

The solution is the pair that gives sum -64.

$$a=-48 b=-16$$

Rewrite 3x^{2}-64x+256 as \left(3x^{2}-48x\right)+\left(-16x+256\right).

$$\left(3x^{2}-48x\right)+\left(-16x+256\right)$$

Factor out 3x in the first and -16 in the second group.

$$3x\left(x-16\right)-16\left(x-16\right)$$

Factor out common term x-16 by using distributive property.

$$\left(x-16\right)\left(3x-16\right)$$

Rewrite the complete factored expression.

$$\frac{\left(x-16\right)\left(3x-16\right)}{4}$$

Evaluate

$$\frac{\left(x-16\right)\left(3x-16\right)}{4}$$