To factor the quadratic expression, we will start by identifying the values of a, b, and c.
y^2-7y-30 ⇔ 1y^2+( - 7)y+( - 30)
For our expression, we have that a= 1, b= - 7, and c= - 30.
To factor a quadratic expression with leading coefficient a= 1, we need to find two factors of c= - 30 whose sum is b= - 7. Since - 30 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 - 30
^(1* (- 30)) - 30
1+(- 30) - 29
- 1 and 30
^(- 1* 30) - 30
^(- 1+30) 29
2 and - 15
^(2* (- 15)) - 30
^(2+(- 15)) - 13
- 2 and 15
^(- 2* 15) - 30
^(- 2+15) 13
3 and - 10
^(3* (- 10)) - 30
^(3+(- 10)) - 7
- 3 and 10
^(- 3* 10) - 30
^(- 3+10) 7
5 and - 6
^(5* (- 6)) - 30
^(5+(- 6)) - 1
- 5 and 6
^(- 5* 6) - 30
^(- 5+6) 1
The integers whose product is - 30 and whose sum is - 7 are 3 and -10.
y^2-7y-30 ⇔ (y+3)(y-10)
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.
