Can someone please explain the way to solve the problem:
In the xy-plane, at which 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).
(c)
Great thanks!
plane geometry
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Hey kuzzden - Let me help you.
(x+a)(x+b) =0
There can be multiple combinations of a and b that can give multiple points of instersection
For instance, a =-1 , b =0 or a =200 , b = -201 or a = 1000 , b = -1001
So point of intersection at X axis can be (0,0) and (-1,0)
So point of intersection at X axis can be (200,0) and (-201,0)
So point of intersection at X axis can be (1000,0) and (-1001,0)
Hence Insufficient
-6 = ab (as x= 0)
Again a and b can have multiple values
a = 6, b = -1
So point of intersection at X axis can be (6,0) and (-1,0)
a = -2 , b = 3
So point of intersection at X axis can be (3,0) and (-2,0)
Hence Insufficient
Combining both will give us two equations
a+b = -1
ab =6
On solving this we get two values -2 and 3
a = -3 , b = 2
a = 2 , b = -3
You might feel this is wrong too but if you read question stem carefully
So whatever may be the values of a and b point of intersection will be the same in this case (-3,0) and (2,0)
Hence (c)
Hope this helps!
Can someone please explain the way to solve the problem:
Graph will intersect X axis when y = 0In the xy-plane, at which two points does the graph of y = (x+a)(x+b)intersect the x-axis?
Plug y = 0 as asked in question(1) a + b = -1
(x+a)(x+b) =0
There can be multiple combinations of a and b that can give multiple points of instersection
For instance, a =-1 , b =0 or a =200 , b = -201 or a = 1000 , b = -1001
So point of intersection at X axis can be (0,0) and (-1,0)
So point of intersection at X axis can be (200,0) and (-201,0)
So point of intersection at X axis can be (1000,0) and (-1001,0)
Hence Insufficient
Intersects the y-axis at (0,-6) MEANS x =0 and y = -6. Put the values(2) The graph intersects the y-axis at (0,-6).
-6 = ab (as x= 0)
Again a and b can have multiple values
a = 6, b = -1
So point of intersection at X axis can be (6,0) and (-1,0)
a = -2 , b = 3
So point of intersection at X axis can be (3,0) and (-2,0)
Hence Insufficient
Combining both will give us two equations
a+b = -1
ab =6
On solving this we get two values -2 and 3
a = -3 , b = 2
a = 2 , b = -3
You might feel this is wrong too but if you read question stem carefully
you will get to know it asks for two points and not for the values of a and b.at which two points does the graph of y = (x+a)(x+b)intersect the x-axis?
So whatever may be the values of a and b point of intersection will be the same in this case (-3,0) and (2,0)
Hence (c)
Hope this helps!
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When posting questions, please use the spoiler function to hide the correct answer. This will allow others to attempt the question without seeing the final answer.kuzzden wrote:Can someone please explain the way to solve the problem:
In the xy-plane, at which 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).
Great thanks!
Target question: At which two points of the graph does y=(x+a)(x+b) intersect the x-axis?
IMPORTANT: Let's take a close look at 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)
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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haha ..Cmon Dude!! :mrgreen:
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