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# If f(x)=ax²+bx+c, for all x is f(x)<0?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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If f(x)=ax²+bx+c, for all x is f(x)<0?  [#permalink]

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12 Jun 2017, 01:03
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Difficulty:

65% (hard)

Question Stats:

43% (01:17) correct 57% (01:25) wrong based on 81 sessions

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If f(x)=ax²+bx+c, for all x is f(x)<0?

1) b²-4ac<0
2) a<0

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"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" ##### Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 54496 Re: If f(x)=ax²+bx+c, for all x is f(x)<0? [#permalink] ### Show Tags 16 Jun 2017, 11:39 1 4 MathRevolution wrote: If f(x)=ax²+bx+c, for all x is f(x)<0? 1) b²-4ac<0 2) a<0 Please find below theory that should help with this question: Parabola A parabola is the graph associated with a quadratic function, i.e. a function of the form $$y=ax^2+bx+c$$. The general or standard form of a quadratic function is $$y =ax^2+bx+c$$, or in function form, $$f(x)=ax^2+bx+c$$, where $$x$$ is the independent variable, $$y$$ is the dependent variable, and $$a$$, $$b$$, and $$c$$ are constants. • If $$a$$ is positive, the parabola opens upward, if negative, the parabola opens downward. x-intercepts: The x-intercepts, if any, are also called the roots of the function. The x-intercepts are the solutions to the equation $$0=ax^2+bx+c$$ and can be calculated by the formula: $$x_1=\frac{-b-\sqrt{b^2-4ac}}{2a}$$ and $$x_2=\frac{-b+\sqrt{b^2-4ac}}{2a}$$ Expression $$b^2-4ac$$ is called discriminant: • If discriminant is positive parabola has two intercepts with x-axis; • If discriminant is negative parabola has no intercepts with x-axis; • If discriminant is zero parabola has one intercept with x-axis (tangent point). BACK TO THE QUESTION: If f(x)=ax²+bx+c, for all x is f(x)<0? According to the theory presented above, the question asks whether entire parabola is below x-axis. This will happen if discriminant is negative AND the parabola opens downward. (1) b²-4ac<0 --> the discriminant is negative. Not sufficient. (2) a<0 --> the parabola opens downward. Not sufficient. (1)+(2) Both conditions satisfied. Sufficient. Answer: C. For more check here: https://gmatclub.com/forum/math-coordin ... 87652.html Hope it helps. _________________ ##### General Discussion Senior Manager Joined: 06 Jul 2016 Posts: 364 Location: Singapore Concentration: Strategy, Finance Re: If f(x)=ax²+bx+c, for all x is f(x)<0? [#permalink] ### Show Tags 12 Jun 2017, 09:01 2 MathRevolution wrote: If f(x)=ax²+bx+c, for all x is f(x)<0? 1) b²-4ac<0 2) a<0 S1 -> b²-4ac<0 => a and c are the same sign, can be negative and can be positive. If a & c are positive, f(x) > 0, but if a & c are negative, f(x)<0 Insufficient. S2 -> a<0 No information about c => insufficient. S1+S2 a & c are the same sign and a<0 which implies c<0 Therefore, f(x)<0 for any value of x. C is the answer IMO. _________________ Put in the work, and that dream score is yours! SVP Joined: 26 Mar 2013 Posts: 2137 Re: If f(x)=ax²+bx+c, for all x is f(x)<0? [#permalink] ### Show Tags 12 Jun 2017, 10:41 MathRevolution wrote: If f(x)=ax²+bx+c, for all x is f(x)<0? 1) b²-4ac<0 2) a<0 Dear GMATPrepNow, Can you please share your thoughts about the math behind this question? As I know, when the discriminate is negative, there is no real solution exits. Thanks in advance Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7245 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If f(x)=ax²+bx+c, for all x is f(x)<0? [#permalink] ### Show Tags 14 Jun 2017, 01:25 ==> In the original condition, there are 3 variables (a, b, c) and in order to match the number of variables to the number of equations, there must be 3 equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), if discriminant =b^2-4ac<0, it doesn’t meet with the x-axis, and if a<0, you always get f(x)<0, hence yes, it is sufficient. Therefore, the answer is C. Answer: C _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
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If f(x)=ax²+bx+c, for all x is f(x)<0?  [#permalink]

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14 Jun 2017, 07:33
If b2-4ac<0 then it is complex number. Also we don't know the value of b here. Can someone explain in detail the logic behind in this question
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Re: If f(x)=ax²+bx+c, for all x is f(x)<0?  [#permalink]

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21 Mar 2019, 16:47
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Re: If f(x)=ax²+bx+c, for all x is f(x)<0?   [#permalink] 21 Mar 2019, 16:47
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