Method 1: Factoring
m^2 + 7m + 10
We need to find two numbers that multiply to 10 and add up to 7. These numbers are 2 and 5.
m^2 + 2m + 5m + 10
(m + 2)(m + 5)
n^2 + 9n + 18
We need to find two numbers that multiply to 18 and add up to 9. These numbers are 3 and 6.
n^2 + 3n + 6n + 18
(n + 3)(n + 6)
p^2 - 6p + 8
We need to find two numbers that multiply to 8 and add up to -6. There are no such numbers, so we cannot factor this quadratic.
Method 2: Quadratic Formula
m^2 + 7m + 10
a = 1, b = 7, c = 10
m = (-b ± sqrt(b^2 - 4ac)) / 2a
m = (-7 ± sqrt(7^2 - 4(1)(10))) / 2(1)
m = (-7 ± sqrt(9)) / 2
m = -5 or -2
n^2 + 9n + 18
a = 1, b = 9, c = 18
n = (-b ± sqrt(b^2 - 4ac)) / 2a
n = (-9 ± sqrt(9^2 - 4(1)(18))) / 2(1)
n = (-9 ± sqrt(9)) / 2
n = -3 or -6
p^2 - 6p + 8
a = 1, b = -6, c = 8
p = (-b ± sqrt(b^2 - 4ac)) / 2a
p = (6 ± sqrt(6^2 - 4(1)(8))) / 2(1)
p = (6 ± sqrt(16)) / 2
p = 2 or 4
Therefore, the solutions to the quadratics are:
m^2 + 7m + 10 = (m + 2)(m + 5) or m = -5 or -2
n^2 + 9n + 18 = (n + 3)(n + 6) or n = -3 or -6
p^2 - 6p + 8 = (p - 2)(p - 4) or p = 2 or 4