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X-intercepts of the function \(f(x)\) or in our case the function (graph) \(y=(x+a)(x+b)\) is the value(s) of \(x\) for \(y=0\). So basically the question asks to find the roots of quadratic equation \((x+a)(x+b)=0\).
Statement (1) gives the value of \(a+b\), but we don't know the value of \(ab\) to solve the equation.
Statement (2) tells us the point of y-intercept, or the value of \(y\) when \(x=0\) --> \(y=(x+a)(x+b)=(0+a)(0+b)=ab=-6\). We know the value of \(ab\) but we don't know the value of \(a+b\) to solve the equation.
Together we know the values of both \(a+b\) and \(ab\), hence we can solve the quadratic equation, which will be the x-intercepts of the given graph.
For more on this topic check Coordinate Geometry chapter of Math Book (link in my signature).
Re: GMAT Prep Question, Any Shortcuts? [#permalink]
24 Dec 2009, 10:37
y=(x+a)(x+b) when y=0 To solve this one, what do we need to know? Obviously a or b, which are not stated in the information (1) & (2) So one rule advise by MGMAT Book, always expand when the information given is factorized or Factorized when the information given is expended. We know from a quadratic expression, the x axis intersect when y=0 So let's expand, and we have x^2 + (a + b)x + ab = 0 So now, we need to know ab and (a+b) to solve this equation Therefore the correct answer is C, since only both information taken together permit to answer the question.
At what two points does the graph of y = (x+a)(x+b) intersect the x axis?
You don't need to worry what the equation represents. Just think, what does 'intersection with x axis' imply? It means the y co-ordinate is 0.
0 = (x+a)(x+b) or x = -a or -b Hence the graph must intersect the x axis at points (-a, 0) and (-b, 0). We need the values of a and b now.
Statement 1: a + b = -1 Two variables, only one equation. Not sufficient.
Statement 2: Graph intersects the y axis at (0, -6). At y axis, x = 0. This means when x = 0, y co-ordinate is -6. Put these values in y = (x+a)(x+b) to get -6 = ab. Again, two variables, one equation. Not sufficient alone.
Using both statements, we have two variables and two different equations so we will be able to find the values of a and b. It doesn't matter which is 'a' and which is 'b'. We find that the two of them are -3 and 2. Since we need the points (-a, 0) and (-b, 0), the required points are (3, 0) and (-2, 0). Sufficient.
Started from option 2: ab = -6 Possible values are (2,-3); (3,-2); (-6,1); (1,-6) --> INSUFFICIENT.
Option 1: substitute for x and y in equation we get a+b = -1 Several possible values such as (-3,2); (-8,7) and so on. --> INSUFFICIENT.
Combining both --> find from option 2 which possible value leads to a+b =-1, only one of the four choices does that (2,-3). Hence SUFFICIENT. Answer choice C.
This is not a good idea to plug the numbers for this problem, it's better to understand the concept and you won't need any math at all (certainly you won't need to solve a+b=-1 and ab=-6).
Next, there are infinitely many values of a and b possible to satisfy ab=-6, not just four: notice that we are not told that a and b are integers only, so for example a=1/2 and b=-12 is also 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)
So, it's important to know --- when the quadratic is given in factored form --- y= (x + a) (x + b) --- then we know the two roots, x = -a, and x = -b. Roots are the x-intercepts, the places where the graph intersects the x-axis. Basically, the prompt is asking us to find the values of a & b.
Statement #1: a + b = -1
One equation for two unknowns. Not enough to solve. Not sufficient.
Statement #2: The graph intersects the y-axis at (0, -6)
Plugging in x = 0 (the condition of the y-axis), we get y = (0+a)(0+b) = ab = -6
Again, one equation for two unknowns. Not enough to solve. Not sufficient.
Combined statements: a + b = -1 ab = -6
Two equations with two unknowns ---> we can solve for the values of a & b, which will answer the question. Sufficient.
Answer = C
Here's another practice question on quadratics for practice. http://gmat.magoosh.com/questions/120 When you submit your answer to that question, the next page will have a full video explanation.