Solve 45+2a-8a^2

$$45 + 2 a - 8 a ^ { 2 }$$

Factor

$$-\left(2a-5\right)\left(4a+9\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-8a^{2}+2a+45$$

Factor the expression by grouping. First, the expression needs to be rewritten as -8a^{2}+pa+qa+45. To find p and q, set up a system to be solved.

$$p+q=2 pq=-8\times 45=-360$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -360.

$$-1,360 -2,180 -3,120 -4,90 -5,72 -6,60 -8,45 -9,40 -10,36 -12,30 -15,24 -18,20$$

Calculate the sum for each pair.

$$-1+360=359 -2+180=178 -3+120=117 -4+90=86 -5+72=67 -6+60=54 -8+45=37 -9+40=31 -10+36=26 -12+30=18 -15+24=9 -18+20=2$$

The solution is the pair that gives sum 2.

$$p=20 q=-18$$

Rewrite -8a^{2}+2a+45 as \left(-8a^{2}+20a\right)+\left(-18a+45\right).

$$\left(-8a^{2}+20a\right)+\left(-18a+45\right)$$

Factor out -4a in the first and -9 in the second group.

$$-4a\left(2a-5\right)-9\left(2a-5\right)$$

Factor out common term 2a-5 by using distributive property.

$$\left(2a-5\right)\left(-4a-9\right)$$

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.

$$-8a^{2}+2a+45=0$$

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.

$$a=\frac{-2±\sqrt{2^{2}-4\left(-8\right)\times 45}}{2\left(-8\right)}$$

Square 2.

$$a=\frac{-2±\sqrt{4-4\left(-8\right)\times 45}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$a=\frac{-2±\sqrt{4+32\times 45}}{2\left(-8\right)}$$

Multiply 32 times 45.

$$a=\frac{-2±\sqrt{4+1440}}{2\left(-8\right)}$$

Add 4 to 1440.

$$a=\frac{-2±\sqrt{1444}}{2\left(-8\right)}$$

Take the square root of 1444.

$$a=\frac{-2±38}{2\left(-8\right)}$$

Multiply 2 times -8.

$$a=\frac{-2±38}{-16}$$

Now solve the equation a=\frac{-2±38}{-16} when ± is plus. Add -2 to 38.

$$a=\frac{36}{-16}$$

Reduce the fraction \frac{36}{-16} to lowest terms by extracting and canceling out 4.

$$a=-\frac{9}{4}$$

Now solve the equation a=\frac{-2±38}{-16} when ± is minus. Subtract 38 from -2.

$$a=-\frac{40}{-16}$$

Reduce the fraction \frac{-40}{-16} to lowest terms by extracting and canceling out 8.

$$a=\frac{5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{9}{4} for x_{1} and \frac{5}{2} for x_{2}.

$$-8a^{2}+2a+45=-8\left(a-\left(-\frac{9}{4}\right)\right)\left(a-\frac{5}{2}\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-8a^{2}+2a+45=-8\left(a+\frac{9}{4}\right)\left(a-\frac{5}{2}\right)$$

Add \frac{9}{4} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-8a^{2}+2a+45=-8\times \frac{-4a-9}{-4}\left(a-\frac{5}{2}\right)$$

Subtract \frac{5}{2} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-8a^{2}+2a+45=-8\times \frac{-4a-9}{-4}\times \frac{-2a+5}{-2}$$

Multiply \frac{-4a-9}{-4} times \frac{-2a+5}{-2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-8a^{2}+2a+45=-8\times \frac{\left(-4a-9\right)\left(-2a+5\right)}{-4\left(-2\right)}$$

Multiply -4 times -2.

$$-8a^{2}+2a+45=-8\times \frac{\left(-4a-9\right)\left(-2a+5\right)}{8}$$

Cancel out 8, the greatest common factor in -8 and 8.

$$-8a^{2}+2a+45=-\left(-4a-9\right)\left(-2a+5\right)$$

Evaluate

$$45+2a-8a^{2}$$