vinay1983 wrote:In the xy-plane, at what two points does the graph y = (x+a)(x+b) intersect the x-axis?
(1) a + b = −6
(2) The graph contains the point (0,−7).
Target question:
At which two points of the graph does y = (x+a)(x+b) intersect the x-axis?
IMPORTANT: Let's examine the point where a line (or curve) crosses the x-axis. At the point of intersection, the point is on the x-axis, which means that the y-coordinate of that point is
0.
So,
for example, to find the point where the line y = 2x+3 crosses the x-axis, we let y =
0 and solve for x.
We get:
0 = 2x+3
When we solve this for x, we get x = -3/2.
So, the line y=2x+3 crosses the x-axis at (-3/2,
0)
Likewise, to determine the point where y = (x + a)(x + b) crosses the x axis, let y=0 and solve for x.
We get: 0 = (x + a)(x + b), which means x = -a or x = -b
This means that
y = (x + a)(x + b) crosses the x axis at (-a, 0) and (-b, 0)
So, to solve this question,
we need the values of a and b
Aside: y = (x + a)(x + b) is actually a parabola. This explains why it crosses the x axis at two points.
Now let's rephrase the target question:
Rephrased target question:
What are the values of a and b?
Statement 1: a + b = -6
There's no way we can use this to determine the values of a and b.
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: The graph contains the point (0,-7)
So, when x = 0, y = -7
When we plug x = 0 and y = -7 into the given equation, we get -7 = (0 + a)(0 + b), which tells us that ab = -7
So, statement 2 is a fancy way of telling us that ab = -7
Since there's no way we can use this information to determine the values of a and b, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined:
Statement 1 tells us that a+b = -6
Statement 2 tells us that ab = -7
In our heads we can see that there are two possible solutions:
- solution a: a = -7 and b = 1
- solution b: a = 1 and b = -7
So, the original equation is either y = (x - 7)(x + 1) or y = (x + 1)(x - 7). Of course, these two equations are the same, and they're both such that
the two points of intersection are (7, 0) and (-1, 0)
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer =
C
Cheers,
Brent
