Multiple choice

If $p, q, r$ are real and $\displaystyle p\neq q,$ then roots of the equation $\displaystyle \left ( p-q \right )x^{2}+5\left ( p+q \right )x-2\left ( p-q \right )=0$ are

  1. Real and equal

  2. Complex

  3. Real and unequal

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The discriminant is 25(p+q)^2 + 8(p-q)^2, which is strictly positive because p-q is nonzero. Therefore, the quadratic has two real and unequal roots.

AI explanation

To determine the nature of the roots, we calculate the discriminant D = b^2 - 4ac using the coefficients from the given equation. Substituting the values gives D = [5(p+q)]^2 - 4(p-q)(-2(p-q)), which simplifies to 25(p+q)^2 + 8(p-q)^2. Since p and q are real and p does not equal q, the term 8(p-q)^2 is strictly positive, making the entire discriminant greater than zero. Therefore, the roots of the equation are real and unequal.

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