Answer:
x = -3, x = -9
Explanation:
The x-intercepts of a parabola are the points where the graph crosses the x-axis.
Application
For the function y = x²+12x+27, the x-intercepts are the x-values that are solutions to y = 0. For each such x-value x=p, the function has a factor (x-p). That means we can find the x-intercepts by putting the equation into factored form.
Factored form
The factored form of this quadratic can be written as ...
y = (x +p)(x +q)
Expanding that, we can compare to the original equation:
y = x² +(p+q)x +pq
The values of p and q will satisfy the relations ...
For integer values of p and q, it generally works well to list the ways the product 27 can be formed. The divisors of 27 are 1, 3, 9, 27, so the number of ways they can be combined is limited:
27 = 1·27 = 3·9
We note that the sums of these factors are 28 and 12, so we're interested in the factors 3 and 9, whose sum is 12. That is ...
y = x² +12x +27 = (x +3)(x +9)
Zero Product Rule
The zero product rule tells us the value of y will be zero if and only if at least one of the factors is zero. The x-values that make that true are ...
x +3 = 0 ⇒ x = -3
x +9 = 0 ⇒ x = -9
The x-intercepts of the function are x = -9 and x = -3.