Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
Does y=ax^2+bx+c intersect the x-axis?
1) a<0
2) c>0
\[?\,\,\,\,:\,\,\,\,a{x^2} + bx + c = 0\,\,\,\,{\text{has}}\,\,\left( {{\text{real}}} \right)\,\,{\text{roots?}}\]
\[\left( 1 \right)\,\,\,a < 0\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( { - 1,0,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{Yes}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\,\left[ {y = - {x^2}\,\,\,{\text{parabola}}} \right] \hfill \\
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( { - 1,0, - 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{No}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\,\left[ {y = - {x^2} - 1\,\,\,{\text{parabola}}} \right] \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,c > 0\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {0,1,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{Yes}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\,\left[ {y = x + 1\,\,{\text{line}}} \right] \hfill \\
\,{\text{Take}}\,\,\left( {a,b,c} \right) = \left( {1,0,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{No}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\,\left[ {y = {x^2} + 1\,\,\,{\text{parabola}}} \right] \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\,a \ne 0\,\,\,\,\, \Rightarrow \,\,\,\,\,?\,\,\,\,:\,\,\,\,\Delta = {b^2} - 4ac\,\,\,\mathop \geqslant \limits^? \,\,\,0\]
\[\left. \begin{gathered}
a < 0\,\, \hfill \\
c > 0 \hfill \\
\end{gathered} \right\}\,\,\,\,\, \Rightarrow \,\,\,\, - 4ac > 0\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{b^2}\,\, \geqslant \,\,0} \,\,\,\,\Delta > 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUF}}.\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
