To factor the quadratic expression, we will start by identifying the values of a, b, and c.
a^2+8a-48 ⇔ 1a^2+ 8a+( - 48)
For our expression, we have that a= 1, b= 8, and c= - 48.
To factor a quadratic expression with leading coefficient a= 1, we need to find two factors of c= - 48 whose sum is b= 8. Since - 48 is a negative number, we will only consider factors with opposite signs — one positive and one negative — so that their product is negative.
Factor Pair
Product
Sum
1 and - 48
^(1* (- 48)) - 48
1+(- 48) - 47
- 1 and 48
^(- 1* 48) - 48
^(- 1+48) 47
2 and - 24
^(2* (- 24)) - 48
^(2+(- 24)) - 22
- 2 and 24
^(- 2* 24) - 48
^(- 2+24) 22
3 and - 16
^(3* (- 16)) - 48
^(3+(- 16)) - 13
- 3 and 16
^(- 3* 16) - 48
^(- 3+16) 13
4 and - 12
^(4* (- 12)) - 48
^(4+(- 12)) - 8
- 4 and 12
^(- 4* 12) - 48
^(- 4+12) 8
6 and - 8
^(6* (- 8)) - 48
^(6+(- 8)) - 2
- 6 and 8
^(- 6* 8) - 48
^(- 6+8) 2
The integers whose product is - 48 and whose sum is 8 are - 4 and 12.
a^2+8a-48 ⇔ (a-4)(a+12)
Let's use a graphing calculator to confirm our answer. To do so, we will graph the related functions in the same coordinate plane. Note that in the calculator we will use the variable x instead of a.
We see that only one graph appears. This means that both graphs coincide. Therefore, the expression has been factored correctly.
