Solve -1/4x^2+3/8x+27/64

$$- \frac { 1 } { 4 } x ^ { 2 } + \frac { 3 } { 8 } x + \frac { 27 } { 64 }$$

Factor

$$\frac{\left(-4x-3\right)\left(4x-9\right)}{64}$$

Solution Steps

Factor out \frac{1}{64}.

$$\frac{-16x^{2}+24x+27}{64}$$

Consider -16x^{2}+24x+27. Factor the expression by grouping. First, the expression needs to be rewritten as -16x^{2}+ax+bx+27. To find a and b, set up a system to be solved.

$$a+b=24 ab=-16\times 27=-432$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -432.

$$-1,432 -2,216 -3,144 -4,108 -6,72 -8,54 -9,48 -12,36 -16,27 -18,24$$

Calculate the sum for each pair.

$$-1+432=431 -2+216=214 -3+144=141 -4+108=104 -6+72=66 -8+54=46 -9+48=39 -12+36=24 -16+27=11 -18+24=6$$

The solution is the pair that gives sum 24.

$$a=36 b=-12$$

Rewrite -16x^{2}+24x+27 as \left(-16x^{2}+36x\right)+\left(-12x+27\right).

$$\left(-16x^{2}+36x\right)+\left(-12x+27\right)$$

Factor out -4x in the first and -3 in the second group.

$$-4x\left(4x-9\right)-3\left(4x-9\right)$$

Factor out common term 4x-9 by using distributive property.

$$\left(4x-9\right)\left(-4x-3\right)$$

Rewrite the complete factored expression.

$$\frac{\left(4x-9\right)\left(-4x-3\right)}{64}$$

Evaluate

$$-\frac{x^{2}}{4}+\frac{3x}{8}+\frac{27}{64}$$