aeros232 wrote:
If the graph of y = x^2 + ax + b passes through the points (m, 0) and (n, 0), where m < n, what is the value of n – m ?
(1) 4b = a^2 – 4
(2) b = 0
\(? = n - m\)
\(n > m\,\,\,{\rm{are}}\,\,{\rm{the}}\,\,{\rm{roots}}\,\,{\rm{of}}\,\,{\rm{the}}\,\,{\rm{equation}}\,\,\,{x^2} + ax + b = 0\,\,\,\,\left( * \right)\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{sum/product}}} \,\,\,\,\,\,\,\,\left\{ \matrix{
\,m + n = - a \hfill \cr
\,mn = b \hfill \cr} \right.\)
\(\left( 1 \right)\,\,4b = {a^2} - 4\,\,\,\,\, \Rightarrow \,\,\,\,\,\Delta = {a^2} - 4b = 4\)
\(\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{
\,n = {{ - a + \sqrt \Delta } \over 2} = - {a \over 2} + 1 \hfill \cr
\,m = {{ - a - \sqrt \Delta } \over 2} = - {a \over 2} - 1 \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = n - m = 2\)
\(\left( 2 \right)\,\,b = 0\,\,\,\, \Rightarrow \,\,\,\,mn = 0\,\,\,\,\,\)
\(\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {n,m} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\,\,a = - 1\,\,\,\,\, \Rightarrow \,\,\,\,{x^2} - x = 0\,\,\,\,\left( {{\rm{viable}}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 1 \hfill \cr
\,{\rm{Take}}\,\,\left( {n,m} \right) = \left( {2,0} \right)\,\,\,\, \Rightarrow \,\,\,\,a = - 2\,\,\,\,\, \Rightarrow \,\,\,\,{x^2} - 2x = 0\,\,\,\,\left( {{\rm{viable}}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 2\,\, \hfill \cr} \right.\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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