If −5 is a Root of 2x² + px − 15 = 0 and p(x² + x) + k = 0 Has Equal Roots, Find k

Question:

If \( -5 \) is a root of the quadratic equation

\( 2x^2+px-15=0 \)

and the quadratic equation

\( p(x^2+x)+k=0 \)

has equal roots, find the value of \(k\).

Solution

Since \( -5 \) is a root of \( 2x^2+px-15=0 \), substituting \( x=-5 \),

\( 2(-5)^2+p(-5)-15=0 \)

\( 50-5p-15=0 \)

\( 35-5p=0 \)

\( p=7 \)

Substituting \( p=7 \) in the second equation,

\( 7x^2+7x+k=0 \)

For equal roots,

\( D=b^2-4ac=0 \)

\( 7^2-4(7)(k)=0 \)

\( 49-28k=0 \)

\( k=\frac{49}{28} \)

\( k=\frac{7}{4} \)

Answer

\( \boxed{k=\frac{7}{4}} \)