Solve f(x)=-8/225x^2+x+2

$$f ( x ) = - \frac { 8 } { 225 } x ^ { 2 } + x + 2$$

Factor

$$\frac{\left(30-x\right)\left(8x+15\right)}{225}$$

Solution Steps

Factor out \frac{1}{225}.

$$\frac{-8x^{2}+225x+450}{225}$$

Consider -8x^{2}+225x+450. Factor the expression by grouping. First, the expression needs to be rewritten as -8x^{2}+ax+bx+450. To find a and b, set up a system to be solved.

$$a+b=225 ab=-8\times 450=-3600$$

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 -3600.

$$-1,3600 -2,1800 -3,1200 -4,900 -5,720 -6,600 -8,450 -9,400 -10,360 -12,300 -15,240 -16,225 -18,200 -20,180 -24,150 -25,144 -30,120 -36,100 -40,90 -45,80 -48,75 -50,72 -60,60$$

Calculate the sum for each pair.

$$-1+3600=3599 -2+1800=1798 -3+1200=1197 -4+900=896 -5+720=715 -6+600=594 -8+450=442 -9+400=391 -10+360=350 -12+300=288 -15+240=225 -16+225=209 -18+200=182 -20+180=160 -24+150=126 -25+144=119 -30+120=90 -36+100=64 -40+90=50 -45+80=35 -48+75=27 -50+72=22 -60+60=0$$

The solution is the pair that gives sum 225.

$$a=240 b=-15$$

Rewrite -8x^{2}+225x+450 as \left(-8x^{2}+240x\right)+\left(-15x+450\right).

$$\left(-8x^{2}+240x\right)+\left(-15x+450\right)$$

Factor out 8x in the first and 15 in the second group.

$$8x\left(-x+30\right)+15\left(-x+30\right)$$

Factor out common term -x+30 by using distributive property.

$$\left(-x+30\right)\left(8x+15\right)$$

Rewrite the complete factored expression.

$$\frac{\left(-x+30\right)\left(8x+15\right)}{225}$$

Evaluate

$$-\frac{8x^{2}}{225}+x+2$$