Solve 33+8a-a^2

$$33 + 8 a - a ^ { 2 }$$

Factor

$$-\left(a-11\right)\left(a+3\right)$$

Solution Steps

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

$$-a^{2}+8a+33$$

Factor the expression by grouping. First, the expression needs to be rewritten as -a^{2}+pa+qa+33. To find p and q, set up a system to be solved.

$$p+q=8 pq=-33=-33$$

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 -33.

$$-1,33 -3,11$$

Calculate the sum for each pair.

$$-1+33=32 -3+11=8$$

The solution is the pair that gives sum 8.

$$p=11 q=-3$$

Rewrite -a^{2}+8a+33 as \left(-a^{2}+11a\right)+\left(-3a+33\right).

$$\left(-a^{2}+11a\right)+\left(-3a+33\right)$$

Factor out -a in the first and -3 in the second group.

$$-a\left(a-11\right)-3\left(a-11\right)$$

Factor out common term a-11 by using distributive property.

$$\left(a-11\right)\left(-a-3\right)$$

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.

$$-a^{2}+8a+33=0$$

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.

$$a=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\times 33}}{2\left(-1\right)}$$

Square 8.

$$a=\frac{-8±\sqrt{64-4\left(-1\right)\times 33}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-8±\sqrt{64+4\times 33}}{2\left(-1\right)}$$

Multiply 4 times 33.

$$a=\frac{-8±\sqrt{64+132}}{2\left(-1\right)}$$

Add 64 to 132.

$$a=\frac{-8±\sqrt{196}}{2\left(-1\right)}$$

Take the square root of 196.

$$a=\frac{-8±14}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-8±14}{-2}$$

Now solve the equation a=\frac{-8±14}{-2} when ± is plus. Add -8 to 14.

$$a=\frac{6}{-2}$$

Divide 6 by -2.

$$a=-3$$

Now solve the equation a=\frac{-8±14}{-2} when ± is minus. Subtract 14 from -8.

$$a=-\frac{22}{-2}$$

Divide -22 by -2.

$$a=11$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and 11 for x_{2}.

$$-a^{2}+8a+33=-\left(a-\left(-3\right)\right)\left(a-11\right)$$

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

$$-a^{2}+8a+33=-\left(a+3\right)\left(a-11\right)$$

Evaluate

$$-\left(a-11\right)\left(a+3\right)$$