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)
In the xy plane..
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- rommysingh
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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
- rommysingh
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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.
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.
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Sorry, I'm not sure what you're asking.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.
Cheers,
Brent
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.
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 we have two variables in the original condition (a,b), we need two equations to solve the problem thus C is likely the answer. Putting 1) & 2) together, y=x^2+(a+b)x+ab and in case of (2) the y-intercept is -6<0, forming a concave parabola that meets the x-axis. Thus the data is sufficient. In case of (1), since we have both a and b value and the parabola meets with the x-axis in x = -a, -b, the data is sufficient. Therefore the best answer is D.
If you know our own innovative logics to find the answer, you don't need to actually solve the problem.
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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 we have two variables in the original condition (a,b), we need two equations to solve the problem thus C is likely the answer. Putting 1) & 2) together, y=x^2+(a+b)x+ab and in case of (2) the y-intercept is -6<0, forming a concave parabola that meets the x-axis. Thus the data is sufficient. In case of (1), since we have both a and b value and the parabola meets with the x-axis in x = -a, -b, the data is sufficient. Therefore the best answer is D.
If you know our own innovative logics to find the answer, you don't need to actually solve the problem.
www.mathrevolution.com
l The one-and-only World's First Variable Approach for DS and IVY Approach for PS that allow anyone to easily solve GMAT math questions.
l The easy-to-use solutions. Math skills are totally irrelevant. Forget conventional ways of solving math questions.
l The most effective time management for GMAT math to date allowing you to solve 37 questions with 10 minutes to spare
l Hitting a score of 45 is very easy and points and 49-51 is also doable.
l Unlimited Access to over 120 free video lessons at https://www.mathrevolution.com/gmat/lesson
l Our advertising video at https://www.youtube.com/watch?v=R_Fki3_2vO8
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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
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