if - 5 is a root of the quadratic equation 2x2+px-15=0 and the quadratic equation p (x^2+x) + k=0 has equal roots. find the value of k

k = 1.75

Step-by-step explanation:

Given that x = - 5 is a root of the equation, then this value makes the equation true. Substitute x = - 5 into the equation and solve for p

2 ( - 5) ² + p ( - 5) - 15 = 0

50 - 5p - 15 = 0

- 5p + 35 = 0 (subtract 35 from both sides)

- 5p = - 35 (divide both sides by - 5)

p = 7

Thus

7 (x² + x) + k = 0, that is

7x² + 7x + k = 0 ← in standard form

Using the discriminant Δ = b² - 4ac to find k

For equal roots then b² - 4ac = 0

with a = 7, b = 7 and c = k, then

7² - (4 * 7 * k) = 0

49 - 28k = 0 (subtract 49 from both sides)

- 28k = - 49 (divide both sides by - 28)

k = 1.75