To factor the quadratic expression, we will start by identifying the values of a, b, and c.
y^2-17y+72 ⇔ 1y^2+( - 17)y+ 72
For our expression, we have that a= 1, b= - 17, and c= 72.
To factor a quadratic expression with leading coefficient a= 1, we need to find two factors of c= 72 whose sum is b= - 17. Since 72 is a positive number, we will only consider factors with the same sign — both positive or both negative — so that their product is positive.
Factor Pair
Product
Sum
1 and 72
^(1* 72) 72
1+72 73
- 1 and - 72
^(- 1* (- 72)) 72
^(- 1+(- 72)) - 73
2 and 36
^(2* 36) 72
^(2+36) 38
- 2 and - 36
^(- 2* (- 36)) 72
^(- 2+(- 36)) - 38
3 and 24
^(3* 24) 72
^(3+24) 27
- 3 and - 24
^(- 3* (- 24)) 72
^(- 3+(- 24)) - 27
4 and 18
^(4* 18) 72
^(4+18) 22
- 4 and - 18
^(- 4* (- 18)) 72
^(- 4+(- 18)) - 22
6 and 12
^(6* 12) 72
^(6+12) 18
- 6 and - 12
^(- 6* (- 12)) 72
^(- 6+(- 12)) - 18
8 and 9
^(8* 9) 72
^(8+9) 17
- 8 and - 9
^(- 8* (- 9)) 72
^(- 8+(- 9)) - 17
The integers whose product is 72 and whose sum is - 17 are - 9 and -8.
y^2-17y+72 ⇔ (y-9)(y-8)
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 y.
We see that only one graph appears. This means that both graphs coincide. Therefore, the expression has been factored correctly.
