To factor the quadratic expression, we will start by identifying the values of a, b, and c.
- 42-m+m^2 ⇔ 1m^2+( - 1)m+( - 42)
For our expression, we have that a= 1, b= - 1, and c= - 42.
To factor a quadratic expression with leading coefficient a= 1, we need to find two factors of c= - 42 whose sum is b= - 1.
Since - 42 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 - 42
^(1* (- 42)) - 42
1+(- 42) - 41
- 1 and 42
^(- 1* 42) - 42
^(- 1+42) - 41
2 and - 21
^(2* (- 21)) - 42
^(2+(- 21)) - 19
- 2 and 21
^(- 2* 21) - 42
^(- 2+21) 19
3 and - 14
^(3* (- 14)) - 42
^(3+(- 14)) - 11
- 3 and 14
^(- 3* 14) - 42
^(- 3+14) 11
6 and - 7
^(6* (- 7)) - 42
^(6+(- 7)) - 1
- 6 and 7
^(- 6* 7) - 42
^(- 6+7) 1
The integers whose product is - 42 and whose sum is - 1 are 6 and -7.
- 42-m+m^2 ⇔ (m+6)(m-7)
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 m.
We see that only one graph appears. This means that both graphs coincide. Therefore, the expression has been factored correctly. ✓
