misterJJ2u wrote:
In the xy-plane, at what two points does the graph of y = (x + a)(x + b) intersect the x-axis?
(1) a + b = -1
(2) The graph intersects the y-axis at (0, -6)
Solution:The two points where the graph of y = (x + a)(x + b) intersects the x-axis are the two x-intercepts of the graph. To determine the x-intercepts, we solve for x when y = 0:
0 = (x + a)(x + b)
x = -a or x = -b
We see that if we can determine the values of a and b, then we can determine the two x-intercepts, namely, -a and -b.
Statement One Alone:
a + b = -1
Since there are many pairs of values that can add to -1, we can’t determine the exact values of a and b. Statement one alone is not sufficient.
Statement Two Alone:The graph intersects the y-axis at (0, -6).
This means when x = 0, y = -6. That is:
-6 = (0 + a)(0 + b)
-6 = ab
Like statement one, there are many pairs of values that can multiply to -6, so we can’t determine the specific values of a and b. Statement two alone is not sufficient.
Statements One and Two Together:From the two statements, we see that a + b = -1 and ab = -6. If we let b = -6/a and substitute it into the first equation, we have:
a + (-6/a) = -1
a^2 - 6 = -a
a^2 + a - 6 = 0
(a + 3)(a - 2) = 0
a = -3 or a = 2
Although it seems there are two distinct values for a instead of one unique value, let’s look at the corresponding values of b also.
Case 1: If a = -3, b = -6/(-3) = 2.
Case 2: If a = 2, b = -6/2 = -3.
Recall that we are looking for the two x-intercepts, which have the values of -a and -b. In the first case, we have the two x-intercepts as 3 and -2 and in the second, we also have the same two x-intercepts, -2 and 3. Therefore, both statements together are sufficient to answer the question.
Answer: C _________________
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