Multiple choice

If $p, q, r$ & $s$ are real numbers & $pr = 2\left( q + s \right)$, then which of the following statement is correct about these equations ${ x }^{ 2 } + px + q = 0, { x }^{ 2 } + rx + s = 0$

  1. Both equations must have real roots

  2. Atleast one of the equation must have real roots

  3. Both equations must have imaginary roots

  4. Exactly one of the equation must have real roots
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Discriminants are D1 = p^2 - 4q and D2 = r^2 - 4s. Sum = D1 + D2 = p^2 + r^2 - 4(q + s). Given pr = 2(q + s), then 2pr = 4(q + s). So D1 + D2 = p^2 + r^2 - 2pr = (p - r)^2. Since the sum of discriminants is >= 0, at least one must be >= 0.

AI explanation

Let D1 = p^2 - 4q and D2 = r^2 - 4s be the discriminants of the two given quadratic equations. From the condition pr = 2(q + s), we can write 4q + 4s = 2pr. Substituting this into the sum of the discriminants gives D1 + D2 = p^2 + r^2 - 4q - 4s = p^2 + r^2 - 2pr = (p - r)^2. Since (p - r)^2 is always greater than or equal to zero for real numbers, the sum of the two discriminants is non-negative, meaning they cannot both be negative. Therefore, at least one of the equations must have real roots.

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