If (−5) is a root of the quadratic equation 2x² + px − 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots, then find the values of p and k.

It is given that 2x² + px − 15 = 0 

If (−5) is a root of this quadratic equation, then 

2(−5)² + p(−5) − 15 = 0 

⇒ 2(25) + p(−5) − 15 = 0 

⇒ 50 − 5p − 15 = 0 

⇒ 5p = 35 

⇒ p = 7 … (1) 

Again, it is given that p(x² + x) + k = 0 

⇒ px² + px + k = 0 

On putting p = 7, we get: 

7x² + 7x + k = 0 

If this equation has real roots, then its discriminant must be 0. 

(7)² − 4 (7) (k) = 0 

⇒ 49 − 28k = 0 

$\Rightarrow \mathrm{k}=\frac{49}{28}=\frac{7}{4}$. 

Thus, the values of p and k are 7 and $\frac{7}{4}$ respectively.