Data Sufficiency

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

Thanks George.

I meant to mention that coordinate geometry questions are typically harder than average questions, so don't worry if you haven't mastered them yet.

If you want to focus your efforts on coordinate geometry questions, you can use BTG's tagging feature. For example, here are all of the questions tagged as coordinate geometry questions: https://www.beatthegmat.com/forums/tags/ ... nate-plane
See the left side of that linked page for more tag options

Cheers,
Brent

Hi sapuna,

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.
Which would lead to different solutions.
Fact 1 is INSUFFICIENT

Fact 2: The graph intersect 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

This is a "system" of equations, so we CAN solve it, but we don't actually have to because the solutions would be flip-flops of one another.

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

These provide the SAME solution.
Combined, SUFFICIENT

Final Answer: C

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