Question

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 the value of k is ______.

Options

• `7/4`

• `5/4`

• `3/4`

• `1/4`

Answer 1

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 the value of k is `underlinebb(7/4)`.

Explanation:

1. Solve for p

Since –5 is a root of the equation 2x² + px – 15 = 0, it must satisfy the equation when substituted for x:

2(–5)² + p(–5) – 15 = 0

2(25) – 5p – 15 = 0 

50 – 5p – 15 = 0 

35 – 5p = 0

5p = 35

p = 7

2. Form the second equation

Substitute the value of p = 7 into the second equation p(x² + x) + k = 0:

7(x² + x) + k = 0

7x² + 7x + k = 0

3. Apply the equal roots condition

For a quadratic equation ax² + bx + c = 0 to have equal roots, its discriminant (D = b² – 4ac) must be zero.

Here, a = 7, b = 7 and c = k:

D = (7)² – 4(7)(k) = 0

49 – 28k = 0

28k = 49

`k = 49/28`

`k = 7/4`