Question

If x² + px - 8 = 0 has roots r and s such that r = -2s, then s is equal to A. 2 B. 2 or 4 C. 2 or -2 D. 4​

Answer 1

Answer:

Step-by-step explanation:

x = (-p ± √(p² + 32)) / 2

Given that r = -2s, we can substitute r with -2s in the above equation to get:

s = (-r / 2) = (s / 4) * (-p ± √(p² + 32))

Multiplying both sides by 4s, we get:

4s² = s * (-p ± √(p² + 32))

Simplifying the above equation, we get:

s * (4s ± p) = √(p² + 32) * s

Dividing both sides by s and simplifying, we get:

4s ± p = √(p² + 32)

Squaring both sides of the above equation, we get:

16s² ± 8ps + p² = p² + 32

Simplifying the above equation, we get:

16s² ± 8ps - 32 = 0

Dividing both sides by 8, we get:

2s² ± ps - 4 = 0

Using the quadratic formula, we get:

s = (-p ± √(p² + 32)) / 4

Therefore, the value of s is equal to 2 or -2 12. Hence, the answer is C.

Answer 2

Answer:

Given the quadratic equation \(x^2 + px - 8 = 0\) with roots \(r\) and \(s\), we know that the sum of the roots (\(r + s\)) is equal to the negation of the coefficient of \(x\) (which is \(p\)), and the product of the roots (\(rs\)) is equal to the constant term (which is \(-8\)).

We are also given that \(r = -2s\). We'll use this information to solve for \(s\).

From the relationship \(r = -2s\), we can write:

\[r + s = 0 \quad \text{(1)}\]

\[r \cdot s = -8 \quad \text{(2)}\]

Substituting \(r = -2s\) into equation (1), we get:

\[-2s + s = -s = 0\]

This implies that \(s = 0\), but \(s\) cannot be zero since it's one of the roots of the quadratic equation. Therefore, we must discard \(s = 0\).

Next, we use equation (2) with \(r = -2s\):

\((-2s) \cdot s = -8\)

\(-2s^2 = -8\)

Dividing both sides by \(-2\), we get:

\[s^2 = 4\]

Taking the square root of both sides, we have:

\[s = \pm 2\]

Thus, the possible values for \(s\) are \(2\) or \(-2\). Therefore, the correct answer is option C: \(2\) or \(-2\).