question_answer

If \[5\] is a root of the quadratic equation \[2{{x}^{2}}+px-15=0\] and the quadratic equation \[p({{x}^{2}}+x)k=0\] has equal roots, find the value of k.

Answer:

Given, \[5\] is a root of \[2{{x}^{2}}+px-15=0\]

then,     \[f(-5)=2{{(-5)}^{2}}+p(-5)-15=0\]

\[50-5p-15=0\]

\[35-5p=0\]

\[5p=35\]

\[p=7\]

Now, putting the value of p, in, \[p({{x}^{2}}+x)+k=0\]

we get               \[7{{x}^{2}}+7x+k=0\]

Now,                 \[D={{b}^{2}}-4ac=0\] (\[\because \] has the equal roots)

then,                 \[49-28k=0\]

\[28k=49\]

\[k=\frac{49}{28}=\frac{7}{4}\]