Multiple choice

If difference of roots of the equation x2 – px + q = 0 is 1, the p2 + 4q2 equals

  1. 2q + 3

  2. (1 – 2q)2

  3. (1 + 2q)2

  4. 2q – 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots r1, r2 satisfy r1+r2 = p, r1*r2 = q. Given |r1-r2| = 1. (r1-r2)^2 = 1 => (r1+r2)^2 - 4r1r2 = 1 => p^2 - 4q = 1. Thus p^2 = 1 + 4q. We want p^2 + 4q^2 = 1 + 4q + 4q^2 = (1 + 2q)^2.

AI explanation

Using the formula for the squared difference of the roots of a quadratic equation, (a - b)^2 = D/a^2. Applying this to x^2 - px + q = 0 gives (-p)^2 - 4(1)(q) = 1^2, which simplifies to p^2 - 4q = 1 and means p^2 = 4q + 1. Substituting this into the expression p^2 + 4q^2 yields 4q + 1 + 4q^2, which factors to (2q + 1)^2.

Keep practicing — related questions