Solve 22a-75+a^2

$$22 a - 75 + a ^ { 2 }$$

Factor

$$\left(a-3\right)\left(a+25\right)$$

Solution Steps

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

$$a^{2}+22a-75$$

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-75. To find p and q, set up a system to be solved.

$$p+q=22 pq=1\left(-75\right)=-75$$

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 -75.

$$-1,75 -3,25 -5,15$$

Calculate the sum for each pair.

$$-1+75=74 -3+25=22 -5+15=10$$

The solution is the pair that gives sum 22.

$$p=-3 q=25$$

Rewrite a^{2}+22a-75 as \left(a^{2}-3a\right)+\left(25a-75\right).

$$\left(a^{2}-3a\right)+\left(25a-75\right)$$

Factor out a in the first and 25 in the second group.

$$a\left(a-3\right)+25\left(a-3\right)$$

Factor out common term a-3 by using distributive property.

$$\left(a-3\right)\left(a+25\right)$$

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.

$$a^{2}+22a-75=0$$

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.

$$a=\frac{-22±\sqrt{22^{2}-4\left(-75\right)}}{2}$$

Square 22.

$$a=\frac{-22±\sqrt{484-4\left(-75\right)}}{2}$$

Multiply -4 times -75.

$$a=\frac{-22±\sqrt{484+300}}{2}$$

Add 484 to 300.

$$a=\frac{-22±\sqrt{784}}{2}$$

Take the square root of 784.

$$a=\frac{-22±28}{2}$$

Now solve the equation a=\frac{-22±28}{2} when ± is plus. Add -22 to 28.

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

Divide 6 by 2.

$$a=3$$

Now solve the equation a=\frac{-22±28}{2} when ± is minus. Subtract 28 from -22.

$$a=-\frac{50}{2}$$

Divide -50 by 2.

$$a=-25$$

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 -25 for x_{2}.

$$a^{2}+22a-75=\left(a-3\right)\left(a-\left(-25\right)\right)$$

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

$$a^{2}+22a-75=\left(a-3\right)\left(a+25\right)$$

Evaluate

$$\left(a-3\right)\left(a+25\right)$$