The Discriminant
Difficult Exam question: \((p-1)x^2+4x+(p-5)=0\) has no real roots. Show that \(p\) satisfiys \(p^2-6p+1>0\).
Using discriminant: \(4^2-4(p-1)(p-5)<0\)
\(16-4(p^2-6p+5)<0\)
\(16-4p^2+24p-20<0\)
\(-4-4p^2+24p<0\)
Divide through by \(-4\): \(p^2-6p+1>0\)
Now find all possible values for \(p\)
Find roots:
\(p^2-6p+1=0\)
Roots are \(3+2\sqrt{ 2 }\) and \(3-2\sqrt{ 2 }\)
Draw the graph and show the inequality.