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[ Math Revolution GMAT math practice question] \(x=?\) \(1) x^3+x^2+x=0\) \(2) x=2x\)
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arvind910619 wrote: fskilnik wrote: MathRevolution wrote: [ Math Revolution GMAT math practice question] \(x=?\) \(1) x^3+x^2+x=0\) \(2) x=2x\) \(? = x\) \(\left( 1 \right)\,\,\,x\left( {{x^2} + x + 1} \right) = 0\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \begin{gathered} x = 0 \hfill \\ \,\,\,{\text{OR}} \hfill \\ {x^2} + x + 1 = 0 \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {\, \Rightarrow }\limits^{\left( * \right)} \,\,\,\,\,x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\) \(\left( * \right)\,\,\,\,{x^2} + x + 1 = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\Delta = {\left( 1 \right)^2}  4 \cdot 1 \cdot 1 < 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^2} + x + 1 > 0\,\,\,\,{\text{for}}\,\,{\text{all}}\,\,\,\,x\,\,\,\,\,\,\,\,\,\,\,\) \(\left( 2 \right)\,\,\,x =  2x\,\,\,\,\, \Rightarrow \,\,\,\,3x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\) This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. Sir how did you reach the second statement? If the discriminant is negative then the equation involves complex numbers but how to arrive at the equation x^3+x^2+x>1. Can you please explain this concept with diagram. Thanks in advance . Hi arvind910619! Thank you for your interest in my solution. 01. The expression \({x^3} + {x^2} + x\) may be written in the form \(x\left( {{x^2} + x + 1} \right)\) and this last "factorized version" is better for our purposes, because a product is zero if and only if (at least) one of the factors is zero, therefore: \(x\left( {{x^2} + x + 1} \right) = 0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,x = 0\,\,\,\,{\text{or}}\,\,\,{x^2} + x + 1 = 0\) 02. The equation \({x^2} + x + 1 = 0\) has no real solutions, because its discriminant ("delta") is negative. Therefore the curve (parabola) does not intercept the horizontal axis. From the fact that the parabola is concave upward (see image attached), we are sure \({x^2} + x + 1\) is always positive, by the way. 03. Finally, from the fact that \(x\left( {{x^2} + x + 1} \right) = 0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,x = 0\,\,\,\,{\text{or}}\,\,\,{x^2} + x + 1 = 0\) and also \({x^2} + x + 1 \ne 0\) for all real values of \(x\), we are sure we must have \(x = 0\) , as mentioned. I hope you got all details! Regards and success in your studies, Fabio. P.S.: If you would like to understand things deeply to be able to perform in the exam in a much higherlevel... try our test drive!
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MathRevolution wrote: [ Math Revolution GMAT math practice question] \(x=?\) \(1) x^3+x^2+x=0\) \(2) x=2x\) Statement 1: \(x^3+x^2+x=0\) \(x(x^2+x+1)=0\) Now the lease value of\(x^2+x+1\) is \(+\frac{3}{4}\) therefore x must be equal to 0 Sufficient.Statement 2: x=2x 3x=0 x=0 Sufficient.Answer: D



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MathRevolution wrote: [ Math Revolution GMAT math practice question] \(x=?\) \(1) x^3+x^2+x=0\) \(2) x=2x\) \(? = x\) \(\left( 1 \right)\,\,\,x\left( {{x^2} + x + 1} \right) = 0\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \begin{gathered} x = 0 \hfill \\ \,\,\,{\text{OR}} \hfill \\ {x^2} + x + 1 = 0 \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {\, \Rightarrow }\limits^{\left( * \right)} \,\,\,\,\,x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\) \(\left( * \right)\,\,\,\,{x^2} + x + 1 = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\Delta = {\left( 1 \right)^2}  4 \cdot 1 \cdot 1 < 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^2} + x + 1 > 0\,\,\,\,{\text{for}}\,\,{\text{all}}\,\,\,\,x\,\,\,\,\,\,\,\,\,\,\,\) \(\left( 2 \right)\,\,\,x =  2x\,\,\,\,\, \Rightarrow \,\,\,\,3x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\) This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio.
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IMO D.
1) This gives us x(x^2 + x + 1) = 0, so either x=0 or x^2 + x + 1. But there is no value of x that fits the equation as x^2 can either be +ve or 0, therefore the only value possible for x=0. Sufficient.
2) The only value that fits the equation is x=0. Sufficient.



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fskilnik wrote: MathRevolution wrote: [ Math Revolution GMAT math practice question] \(x=?\) \(1) x^3+x^2+x=0\) \(2) x=2x\) \(? = x\) \(\left( 1 \right)\,\,\,x\left( {{x^2} + x + 1} \right) = 0\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \begin{gathered} x = 0 \hfill \\ \,\,\,{\text{OR}} \hfill \\ {x^2} + x + 1 = 0 \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {\, \Rightarrow }\limits^{\left( * \right)} \,\,\,\,\,x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\) \(\left( * \right)\,\,\,\,{x^2} + x + 1 = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\Delta = {\left( 1 \right)^2}  4 \cdot 1 \cdot 1 < 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^2} + x + 1 > 0\,\,\,\,{\text{for}}\,\,{\text{all}}\,\,\,\,x\,\,\,\,\,\,\,\,\,\,\,\) \(\left( 2 \right)\,\,\,x =  2x\,\,\,\,\, \Rightarrow \,\,\,\,3x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,x = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\) This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. Sir how did you reach the second statement? If the discriminant is negative then the equation involves complex numbers but how to arrive at the equation x^3+x^2+x>1. Can you please explain this concept with diagram. Thanks in advance .



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=> 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. Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first. Condition 1) \(x^3+x^2+x=0\) \(=> x(x^2+x+1)=0\) \(=> x = 0\) since \(x^2+x+1 ≠ 0\) Condition 1) is sufficient. Condition 2) \(x = 2x\) \(=> 3x = 0\) \(=> x = 0\) Condition 2) is sufficient. Therefore, D is the answer. Answer: D If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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This Question should mention that x is real, then only Statement 1 is sufficient.
Otherwise x = 0 or complex number. It indicates that Statement 1 is not sufficient.










