This is a value question, which means that for a statement(s) to be sufficient you need to get single numerical value of the intersect points.
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Hi Bunuel

I agree. However once we get x=a,b
I used this info in st 1

a+b = -1

Since a=x and b=x

X+x = -1
x= -1/2

I know what I’m doing is not right but theoretically I cannot seem to understand why this approach is incorrect

Thanks

Posted from my mobile device

x-intercepts of a parabola y = (x + a)(x + b) are x = -a and x = -b. This does not mean that -a = -b. For example, The x-intercepts of y = (x + 1)(x + 2) are x = -1 and x = -2.
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In addition to what Bunuel said above, think in this way too:

When we have (x + a)(x + b) = 0, x is a variable and a and b are some constant values.

We know that x must take one of two values: -a or -b
x must be either -a or -b so that the product becomes 0.

When you say a = -x, and b = -x, you are assuming that a and b are equal and their value is some constant -x (which is not the case)

e.g.
(x + 2)(x - 1) = 0 implies x is either -2 or 1. Can I say that -2 = x and 1 = x so (-2 + 1) = x + x = -1?
This gives us x = -1/2 but we know that x is either -2 or 1. Then this is not possible. Our assumption here is that a and b are equal which they needn't be.
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Thankyou Bunuel and VeritasKarishma

It's clear now

I stopped when I had two equation involving A&B thinking A and B will have multiple answer and close E.

For DS do we always need to find complete solution bec looking at the equation one might interpret that this equation would have multiple ans hence information is not sufficient.

Asked: In the xy-plane, at what two points does the graph of y = (x + a)(x + b) intersect the x-axis?

At x = -a & at x = -b, the graph intersect x-axis.

(1) a + b = -1
Multiple values of a & b satisfy the condition.
NOT SUFFICIENT

(2) The graph intersects the y-axis at (0, -6)
At x = 0 ; y=ab = -6
ab = -6
Multiple values of a & b satisfy the condition
NOT SUFFICIENT

(1) + (2)
(1) a + b = -1
(2) The graph intersects the y-axis at (0, -6)
ab = -6
b = -a -1
a(a+1) =6
a = 2 or -3
(a, b) = {(2,-3),(-3,2)}
The graph intersects x-axis at x=3 & x=-2
SUFFICIENT

IMO C
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The general equation of a parabola is y = Ax^2 + Bx + C
The roots of this quadratic equation will give you the points at which the parabola intersects the x-axis.

When you simplify the question stem, you are left with y = x^2 + (a+b)x + ab. So in this case;
A = 1
B = (a+b)
C = (ab)

Hence if you need to find the roots, you need the values of all 3; A, B and C.
S1 gives you the value of B (which is a+b). Not sufficient since you dont have the value of C (which is ab)
S2 gives you the value of C (which is ab). Not sufficient since you dont have the value of B (which is a+b).

When you combine the 2 statements, you have a full parabola equation in the form of y = x^2 - x - 6
Hence your answer is C.

Note: Since its DS, you dont have to solve all the way, but for reference the solution would be (x-3)(x+2); so x= 3 and -2.

Clearly this kind of question is going to be a precious time waster.

Is there a short cut for this type other than just random guessing?

1) a +b = -1
-> a=-2

Insufficient

2) Intersects at (0,-6)
-> a * b = -6
Insufficient

1)+2) = -2*b+-6
b=3

We have value for a = -2 and b=3

This is a quadric equation. If we manipulate it then we get:
\(y=x^2+bx+ax+ab\)

\(y=x^2+x(a+b)+ab\)

To plot the graph we need the value of \((a+b)\) and \(ab\)

(1) Gives the value of (a+b) only INSUFFICIENT

(2) Gives the value of ab only
Putting the values in the equation
\(-6=0^2+0(a+b)+ab\)

\(ab=-6\)

INSUFFICIENT.

Using information from both the option we get values of a+b and ab. SUFFICIENT.

ANS. C
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\(y = x^2 + bx + ax + ab \)
\(y = x^2+ x(a+b) + ab\)

We need the value of a + b and ab.

(1) We don't know the value of ab; INSUFFICIENT.

(2) \(-6 = x^2+ 0(a+b) + ab\)
\(-6 = ab\)
We don't know the value of a+b; INSUFFICIENT.

(1&2) Together, we know ab and a+b; SUFFICIENT.

Answer is C.
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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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Solution:

The equation in question here is (x+a)(x+b)=y

=> x^2 + bx+ ax + ab = y

=> x^2 + x(a+b) + ab = y

This graph would intersect the x-axis at points where x=0. At x=0 we can see y=ab.

St(1)- a + b = -1

There is no information about ab. Hence (Insufficient)

St(2)-The graph intersects the y-axis at (0, -6)

Since at x=0 y is -6,we can substitute the same in y=a b = -6 and have a b=-6

However, with st(2)alone, we have no information of (a+b) . Hence Insufficient

Combining both,

We know (a+b) and (ab).
Sufficient (option c)

Devmitra Sen
GMAT SME





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If the question had asked "AT HOW MANY POINTS DOES THE GRAPH OF y = (x + a)(x + b) INTERSECT X-AXIS?" then "D" would have been the answer. Am I right?