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)
GMATPrep Test 2 Q26
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Strategy:Abhijit K 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)
Since Statement 1 is written in terms of a+b, rephrase the question stem in terms of a+b.
A graph intersects the x-axis when y=0.
Substituting y=0 into y = (x+a)(x+b), we get:
0 = (x+a)(x+b)
0 = x² + ax + bx + ab
0 = x² + (a+b)x + ab.
To determine for which values of x the right-hand side will be equal to 0, we need to know the values of the expressions in red.
Question stem, rephrased:
What are the values of a+b and ab?
Statement 1: a+b = -1
Since the value of ab is unknown, INSUFFICIENT.
Statement 2: The graph intersects the y-axis at (0, -6)
Substituting (0, -6) into y = (x+a)(a+b), we get:
-6 = (0+a)(0+b)
-6 = ab.
Since the value of a+b is unknown, INSUFFICIENT.
Statements combined:
Since a+b = -1 and ab = -6, SUFFICIENT.
The correct answer is C.
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Alternate approach:Abhijit K 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)
The question stem is asking for the x-intercepts.
An x-interceps occurs when y=0.
Statement 1: a+b = -1
Case 1: a=1, b=-2
Substituting a=1 and b=-2 into y= (x+a)(x+b), we get:
y = (x+1)(x-2).
Here, y=0 when x=-1 or x=2.
Thus, the x-intercepts are -1 and 2.
Case 2: a=2, b=-3
Substituting a=2 and b=-3 into y= (x+a)(x+b), we get:
y = (x+2)(x-3).
Here, y=0 when x=-2 or x=3.
Thus, the x-intercepts are -2 and 3.
Since the x-intercepts are different in each case, INSUFFICIENT.
Statement 2: The graph intersects the y-axis at (0, -6)
Substituting (0, -6) into y = (x+a)(a+b), we get:
-6 = (0+a)(0+b)
-6 = ab.
Case 2 also satisfies statement 2, since ab=-6 if a=2 and b=-3.
In Case 2, the x-intercepts are -2 and 3.
Case 3: a=1, b=-6
Substituting a=1 and b=-6 into y= (x+a)(x+b), we get:
y = (x+1)(x-6).
Here, y=0 when x=-1 or x=6.
Thus, the x-intercepts are -1 and 6.
Since the x-intercepts are different in each case, INSUFFICIENT.
Statements combined:
Case 2 satisfies both statements.
In Case 2, the x-intercepts are -2 and 3.
Check whether any other combination for x and y will satisfy both ab=-6 and a+b = -1.
Factor pairs for ab=-6 other than a=2 and b=-3:
a=-6, b=1
a=-3, b=2
a=-2, b=3
a=-1, b=6
a=1, b=-6
a=3, b=-2.
Only the option in red satisfies a+b=-1.
Case 4: a=-3, b=2
Substituting a=-3 and b=2 into y= (x+a)(x+b), we get:
y = (x-3)(x+2).
Here, y=0 when x=3 or x=-2.
Thus, the x-intercepts are -2 and 3.
Since Cases 2 and 4 yield the same x-intercepts, SUFFICIENT.
The correct answer is C.
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Target question: At which two points of the graph does y=(x+a)(x+b) intersect the x-axis?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)
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
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Hi Abhijit K,
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
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