Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
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.
Check out this awesome article about Anderson on Poets Quants, http://poetsandquants.com/2015/01/02/uclas-anderson-school-morphs-into-a-friendly-tech-hub/ . Anderson is a great place! Sorry for the lack of updates recently. I...