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.
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.