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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, then the value of $k$ is:

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\frac{7}{4}

\frac{4}{7}

\frac{2}{7}

\frac{1}{5}

answer is A.

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Detailed Solution

Given quadratic equations are,

2 x^{2} + px - 15 = 0

and

p (x^{2} + x) + k = 0

.
-5 is the root of the equation

2 x^{2} + px - 15 = 0

then,

2 {(- 5)}^{2} + p (- 5) - 15 = 0

⟹ 50 - 5 p - 15 = 0

⟹ 35 - 5 p = 0

⟹ 35 = 5 p

⟹ p = 7

….(1)
And quadratic equation

p (x^{2} + x) + k = 0

has equal roots.
We know that, when roots of the equation are equal then,
Discriminant

(D) = b^{2} - 4 ac = 0

.
Here,

a = p, b = p, c = k

.
Then,

b^{2} - 4 ac = 0

⟹ p^{2} - 4 \times p \times k = 0

⟹ {(7)}^{2} - 4 \times 7 \times k = 0

{Using (1)}

⟹ 49 - 28 k = 0

⟹ 49 = 28 k

⟹ k = \frac{49}{28}

∴ k = \frac{7}{4}

∴ The value of k is

\frac{7}{4}

.
Hence, the correct option is 1.

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