c Compare the results from Part A and Part B. What can you conclude?
A
a x=-3 and x=-2
B
b (x+2)(x+3)
C
c See solution.
Practice makes perfect
a We are given the graph of the quadratic function shown below and asked to determine the x-intercepts. Recall that the x-intercepts are the x-coordinate of the point where the line crosses the x-axis.
As we can see from the graph, the function intersects the x-axis at the points ( -3,0) and ( -2,0). Therefore, the x-intercepts of the function are x = -3 and x = -2.
b Notice that the quadratic function given — y = x^2+5x+6 — is a trinomial of the form x^2+bx+c.
x^2+ bx+ c
x^2+ 5x+ 6To factor a trinomial of this form, we need to find two numbers such that their product is equal to c= 6 and their sum is b= 5. We can analyze the possible combinations using the factors of 6 and organize this information by using a table.
Factors of 6
Sum of Factors
1 and 6
7
2 and 3
0.5cm5 ✓
With this information we can factor the trinomial.
x^2+5x+6 ⇔ (x+2)(x+3)
c In Part B we found the factored form of the quadratic function.
y = x^2+5x+6 ⇔ y = (x+2)(x+3)
Since the x-intercepts are the x-coordinates of the points where the line crosses the x-axis, these happen when y=0. Therefore, we can set the factored form equal to 0 to find them.
(x+2)(x+3) = 0
By the Zero Product Property we can identify the roots of this equation to be x= -2 and x= -3. Notice that these are the same x-values as the x-intercepts of the graph of the function.
