Solve a^2-a-72

$${ a }^{ 2 } -a-72$$

Factor

$$\left(a-9\right)\left(a+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-72. To find p and q, set up a system to be solved.

$$p+q=-1 pq=1\left(-72\right)=-72$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -1.

$$p=-9 q=8$$

Rewrite a^{2}-a-72 as \left(a^{2}-9a\right)+\left(8a-72\right).

$$\left(a^{2}-9a\right)+\left(8a-72\right)$$

Factor out a in the first and 8 in the second group.

$$a\left(a-9\right)+8\left(a-9\right)$$

Factor out common term a-9 by using distributive property.

$$\left(a-9\right)\left(a+8\right)$$

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.

$$a^{2}-a-72=0$$

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.

$$a=\frac{-\left(-1\right)±\sqrt{1-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$a=\frac{-\left(-1\right)±\sqrt{1+288}}{2}$$

Add 1 to 288.

$$a=\frac{-\left(-1\right)±\sqrt{289}}{2}$$

Take the square root of 289.

$$a=\frac{-\left(-1\right)±17}{2}$$

The opposite of -1 is 1.

$$a=\frac{1±17}{2}$$

Now solve the equation a=\frac{1±17}{2} when ± is plus. Add 1 to 17.

$$a=\frac{18}{2}$$

Divide 18 by 2.

$$a=9$$

Now solve the equation a=\frac{1±17}{2} when ± is minus. Subtract 17 from 1.

$$a=-\frac{16}{2}$$

Divide -16 by 2.

$$a=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and -8 for x_{2}.

$$a^{2}-a-72=\left(a-9\right)\left(a-\left(-8\right)\right)$$

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

$$a^{2}-a-72=\left(a-9\right)\left(a+8\right)$$

Evaluate

$$\left(a-9\right)\left(a+8\right)$$