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, then the value of k is:

Q: 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, then the value of k is:

To solve the problem, we need to find the value of $ k $ given the conditions of the quadratic equations. 1. **Condition 1: $-5$ is a root of the equation $2x^2 + px - 15 = 0$.** Since $-5$ is a root, it satisfies the equation: \[ 2(-5)^2 + p(-5) - 15 = 0 \] Simplifying, we get: \[ 2(25) - 5p - 15 = 0 \implies 50 - 5p - 15 = 0 \implies 35 = 5p \implies p = 7 \] 2. **Condition 2: The equation $p(x^2 + x) + k = 0$ has equal roots.** With $p = 7$, the equation becomes: \[ 7(x^2 + x) + k = 0 \implies 7x^2 + 7x + k = 0 \] For the quadratic equation $ax^2 + bx + c = 0$ to have equal roots, the discriminant must be zero. The discriminant $\Delta$ is given by: \[ \Delta = b^2 - 4ac \] Here, $a = 7$, $b = 7$, and $c = k$. Therefore: \[ \Delta = 7^2 - 4 \times 7 \times k = 0 \] Simplifying, we have: \[ 49 - 28k = 0 \implies 49 = 28k \implies k = \frac{49}{28} = \frac{7}{4} \] Thus, the value of $ k $ is $\frac{7}{4}$, which corresponds to Option 3. **Explanation of Incorrect Options:** - **Option 1 (7):** If $ k = 7 $, substituting into the discriminant equation gives: \[ 49 - 28 \times 7 = 49 - 196 \neq 0 \] This does not satisfy the condition for equal roots. - **Option 2 (4):** If $ k = 4 $, substituting into the discriminant equation gives: \[ 49 - 28 \times 4 = 49 - 112 \neq 0 \] This does not satisfy the condition for equal roots. - **Option 4 ($\frac{4}{7}$):** If $ k = \frac{4}{7} $, substituting into the discriminant equation gives: \[ 49 - 28 \times \frac{4}{7} = 49 - 16 \neq 0 \] This does not satisfy the condition for equal roots. Therefore, the only value of $ k $ that satisfies the condition of equal roots is $\frac{7}{4}$, confirming that Option 3 is correct.

Question

Accepted Answer

To solve the problem, we need to find the value of $ k $ given the conditions of the quadratic equations.

1. **Condition 1: $-5$ is a root of the equation $2x^2 + px - 15 = 0$.**

Since $-5$ is a root, it satisfies the equation:
   $$
   2(-5)^2 + p(-5) - 15 = 0
   $$
   Simplifying, we get:
   $$
   2(25) - 5p - 15 = 0 \implies 50 - 5p - 15 = 0 \implies 35 = 5p \implies p = 7
   $$

2. **Condition 2: The equation $p(x^2 + x) + k = 0$ has equal roots.**

With $p = 7$, the equation becomes:
   $$
   7(x^2 + x) + k = 0 \implies 7x^2 + 7x + k = 0
   $$
   For the quadratic equation $ax^2 + bx + c = 0$ to have equal roots, the discriminant must be zero. The discriminant $\Delta$ is given by:
   $$
   \Delta = b^2 - 4ac
   $$
   Here, $a = 7$, $b = 7$, and $c = k$. Therefore:
   $$
   \Delta = 7^2 - 4 \times 7 \times k = 0
   $$
   Simplifying, we have:
   $$
   49 - 28k = 0 \implies 49 = 28k \implies k = \frac{49}{28} = \frac{7}{4}
   $$

Thus, the value of $ k $ is $\frac{7}{4}$, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1 (7):** If $ k = 7 $, substituting into the discriminant equation gives:
  $$
  49 - 28 \times 7 = 49 - 196 \neq 0
  $$
  This does not satisfy the condition for equal roots.

- **Option 2 (4):** If $ k = 4 $, substituting into the discriminant equation gives:
  $$
  49 - 28 \times 4 = 49 - 112 \neq 0
  $$
  This does not satisfy the condition for equal roots.

- **Option 4 ($\frac{4}{7}$):** If $ k = \frac{4}{7} $, substituting into the discriminant equation gives:
  $$
  49 - 28 \times \frac{4}{7} = 49 - 16 \neq 0
  $$
  This does not satisfy the condition for equal roots.

Therefore, the only value of $ k $ that satisfies the condition of equal roots is $\frac{7}{4}$, confirming that Option 3 is correct.