Solve x^2 + 5x + 6 = 0

  x = -2 or -3

Step-by-step explanation:

The given equation can be factored:

  (x +2)(x +3) = 0

The solutions are the values of x that make the factors be zero.

  x + 2 = 0   ⇒   x = -2

  x + 3 = 0   ⇒   x = -3

The solutions are x=-2 or -3.

The product (x +a)(x +b) is ...

  (x +a)(x +b) = x^2 +(a+b)x + ab

Comparing this to the above equation, we see that the product of 'a' and 'b' must be 6, and their sum must be 5. When we look at the factors of 6, we find a couple of choices. The correct one is easy to see.

  6 = 1×6 = 2×3

The sums of factors are 1+6=7 and 2+3=5, so it is the latter pair that is of interest. It does not matter which we call 'a' and which we call 'b'. Once we find two numbers whose sum is 5 and product is 6, we can fill in the binomial factors:

  x^2 +5x +6 = (x +2)(x +3) = 0