Factor form of an equation
Initial explanation
We know that the x intercepts of a function are given when y = 0
like the function of this figure, that intercepts x axis on the blue points.
Then, we want to know the values of x that makes y = 0:
y = 3x² + 18x + 27
↓
0 = 3x² + 18x + 27
Step 1: Great Common Factor
We want to find GCF of the terms of the addition:
0 = 3x² + 18x + 27
Since
3 · 6 = 18 and 3 · 9 = 27, we can write:
0 = 3x² + 18x + 27
↓
0 = 3 · x² + 3 · 6x + 3 · 9
Then, the GCF of the terms of the addition is 3. So we can write:
0 = 3 · x² + 3 · 6x + 3 · 9
↓
0 = 3(x² + 6x + 9)
Step 2: factoring (x² + 6x + 9)
We want to write
x² + 6x + 9 = (x + _ ) (x + _ )
in order to do so we must find the numbers that fill the space.
This couple of numbers satisfy that:
1- their addition is the second term, 6.
2- their product is the last term, 9.
We have that
3 + 3 = 6
and
3 · 3 = 9
Then, the numbers are 3:
x² + 6x + 9 = (x + 3 ) (x + 3) = (x + 3)²
Step 3: finding x- intercepts
Since we have factored completely the equation, we have
3x² + 18x + 27 = 3(x + 3)²
then
3(x + 3)² = 0
We have that this product is zero, 0, iff
(x + 3) = 0
If
x = -3
then
x + 3 = 0
Then
when x = -3, then y = 0
Therefore,
when x = -3, the function intercepts x:
Answer: when x = -3, the function intercepts x.
