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# Does x^2+px+q = 0 have a root?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7597
GMAT 1: 760 Q51 V42
GPA: 3.82
Does x^2+px+q = 0 have a root?  [#permalink]

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29 Aug 2018, 01:54
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55% (hard)

Question Stats:

46% (01:05) correct 54% (00:59) wrong based on 67 sessions

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[Math Revolution GMAT math practice question]

Does $$x^2+px+q = 0$$ have a root?

$$1) p<0$$
$$2) q<0$$

_________________
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 $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" VP Status: Learning stage Joined: 01 Oct 2017 Posts: 1028 WE: Supply Chain Management (Energy and Utilities) Does x^2+px+q = 0 have a root? [#permalink] ### Show Tags 29 Aug 2018, 03:40 MathRevolution wrote: [Math Revolution GMAT math practice question] Does $$x^2+px+q = 0$$ have a root? $$1) p<0$$ $$2) q<0$$ Note:- 1) In GMAT, roots of QE mean REAL roots. 2) A quadratic equation posses real roots when 1) D=0 2) D>0 3) A quadratic equation posses imaginary when D<0 Where D is the discriminant=$$p^2-4*1*q=p^2-4q$$ Polarity of $$p^2-4q$$ is the key decision maker: 1) When p=0, then polarity of D depends on the polarity of q. In order to have obtain real roots, q must be -ve or zer0. (a) Say q=-5, then D=$$p^2-4q$$=0-4*(-5)=20>0, b)say q=0, then D=$$p^2$$-4q=0-4*(0)=0) 2) When p is +ve or -ve, $$p^2$$ is always positive, so, q must be -ve so that D>0. $$1) p<0$$ a) When q>0 and $$p^2>4q$$, roots are imaginary. b) When q=0 or q<0, then D>0. real roots . Insufficient. $$2) q<0$$ Irrespective of sign and value of p, when q<0, $$p^2-4q>0$$ Or, D>0. Hence, roots are real. So QE has roots. sufficient. Ans. (B) _________________ Regards, PKN Rise above the storm, you will find the sunshine Senior Manager Joined: 07 Oct 2017 Posts: 258 Re: Does x^2+px+q = 0 have a root? [#permalink] ### Show Tags 29 Aug 2018, 05:05 MathRevolution wrote: [Math Revolution GMAT math practice question] Does $$x^2+px+q = 0$$ have a root? $$1) p<0$$ $$2) q<0$$ Answer is B Attachment: 1535544405190.jpg [ 62.19 KiB | Viewed 537 times ] Thank you = Kudos _________________ Thank you =Kudos The best thing in life lies on the other side of the pain. Intern Joined: 19 Sep 2013 Posts: 1 Re: Does x^2+px+q = 0 have a root? [#permalink] ### Show Tags 29 Aug 2018, 05:58 Finds a solution on website Finactax. GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 937 Re: Does x^2+px+q = 0 have a root? [#permalink] ### Show Tags 29 Aug 2018, 06:07 MathRevolution wrote: [Math Revolution GMAT math practice question] Does $$x^2+px+q = 0$$ have a root? $$1) p<0$$ $$2) q<0$$ $$?\,\,\,:\,\,\,\,\Delta = \left( {{\text{}}{b^2} - 4ac{\text{}}} \right) = {p^2} - 4q\,\,\,\mathop \geqslant \limits^? \,\,\,\,0\,\,$$ $$\left( 1 \right)\,\,\,\left\{ \begin{gathered} \,Take\,\,\left( {p,q} \right) = \left( { - 1,0} \right)\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\ \,Take\,\,\left( {p,q} \right) = \left( { - 1,1} \right)\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\ \end{gathered} \right.$$ $$\left( 2 \right)\,\,\,q < 0\,\,\,\,\mathop \Rightarrow \limits^{{\text{FOCUS!}}} \,\,\, - 4q > 0\,\,\,\,\mathop \Rightarrow \limits^{{{\text{p}}^{\text{2}}}\,\, \geqslant \,\,0} \,\,\,{p^2} - 4q > 0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle$$ Conclusion: the correct answer is (B). The above follows the notations and rationale taught in the GMATH method. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net VP Status: Learning stage Joined: 01 Oct 2017 Posts: 1028 WE: Supply Chain Management (Energy and Utilities) Re: Does x^2+px+q = 0 have a root? [#permalink] ### Show Tags 29 Aug 2018, 06:08 adityagupta27 wrote: Finds a solution on website Finactax. Hi adityagupta27, Could you please elaborate how Finactax is related to explanation of the question ? _________________ Regards, PKN Rise above the storm, you will find the sunshine Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7597 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Does x^2+px+q = 0 have a root? [#permalink] ### Show Tags 31 Aug 2018, 01:11 => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. The discriminant of the equation is $$p^2 – 4q$$. If the discriminant is greater than or equal to zero, then the quadratic equation has roots. The question asks if $$p^2-4q ≥ 0$$ or not. Since $$p^2 ≥ 0$$, if q < 0, then $$p^2-4q ≥ 0$$. Thus, condition 2) is sufficient. Condition 1) If $$p = -1$$ and $$q = 0$$, then the discriminant is positive and the equation has $$2$$ roots, which are $$0$$ and $$1$$. So, the answer is ‘yes’. If $$p = -1$$ and $$q = 1$$, then the discriminant is negative and the equation has no real roots. So, the answer is ‘no’. Since we don’t have a unique solution, condition 1) is not sufficient. Therefore, B is the answer. Answer: B _________________ 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$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Re: Does x^2+px+q = 0 have a root?   [#permalink] 31 Aug 2018, 01:11
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# Does x^2+px+q = 0 have a root?

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