I solve it and get the answer $a\in [1,7]$.but my teacher told me to take the verification of the boundary values of a.because at the boundary values, $ax^2+3x-4$ and $3x-4x^2+a$ have common roots.so,its obvious that value of a will not count in the answer.But my question is,is it necessary to have common roots between $ax^2+3x-4$ and $3x-4x^2+a$ at only boundary values.if yes then why??
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2Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos Aug 23 '23 at 08:21
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If both roots are common then $a=-4$ // if only one root is common, assume it to be $\alpha$ then solve for $a$ and check – MathStackexchangeIsNotSoBad Aug 23 '23 at 10:28