In the xy-plane, at what 2 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 (0,-6)

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 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 = -1
There's no way we can use this to determine the values of a and b.
Since we can answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The line intercepts the y axis at (0,-6)
This tells us that when x = 0, y = -6
When we plug x = 0 and y = -6 into the equation y = (x + a)(x + b), we get -6 = (0 + a)(0 + b), which tells us that ab=-6
In other words, statement 2 is a fancy way to tell us that ab = -6
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 = -1
Statement 2 tells us that ab = -6

Rewrite equation 1 as a = -1 - b
Then take equation 2 and replace a with (-1 - b) to get: (-1 - b)(b) = -6
Expand: -b - b^2 = -6
Set equal to zero: b^2 + b - 6 = 0
Factor: (b+3)(b-2) = 0
So, b= -3 or b= 2

When b = -3, a = 2 and when b = 2, a = -3
In both cases, the two points of intersection are (3, 0) and (-2, 0)
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

rommysingh wrote:Thanks for a detailed reply Brent. I pretty much followed it all through.
could u explain
"This means that y = (x + a)(x + b) crosses the x axis at (-a, 0) and (-b, 0)"
i know -a,0 and -b,0 but how did we change y= x+a) x+b to the previous one.
rest all is clear.

Sorry, I'm not sure what you're asking.

Cheers,
Brent

Hi rommysingh,

Like most questions on the GMAT, this question can be approached in a number of different ways. There's actually a great Algebra pattern/shortcut built into this question:

We given the equation Y = (X+A)(X+B) and we're asked at what 2 points the graph will intersect with the X-axis. This essentially comes down to the A and B. If we know their values, then we can answer the question. It's also worth noting that since we're multiplying, you can "flip-flop" the values of A and B and you'd have the same solution.

For example: (X+1)(X+2) is the same as (X+2)(X+1)......

Fact 1: A + B = -1

There's no way to determine the exact values for A and B with this information.

TESTing Values, we could have:
A = 0, B = -1
A = 100, B = -101
Etc.
Different numbers for A and B would lead to different solutions.
Fact 1 is INSUFFICIENT

Fact 2: The graph intersects the Y-axis at (0, -6).

Now we have one of the points on the graph. Plugging it into the original equation gives us....

-6 = (0+A)(0+B)
-6 = AB

We have the same situation as in Fact 1: more than 1 possible solution.
A = 1, B = -6
A = 2, B = -3
Etc.
Fact 2 is INSUFFICIENT

Combined, we have...
A + B = -1
AB = -6

Here's where things get interesting. This is a "system" of equations, so we CAN solve it...the "catch" is that the answers would "flip flop":

If you did do the math, you'd have
A = -3, B = 2
OR
A = 2, B = -3

Since this is a graphing question, these two options provide the SAME solution.
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich