Multiple choice

Consider the following statements in respect of the quadratic equation $4(x-p)(x-q)-r^2=0$ Where $p,q$ and $r$ are real numbers : 1.The roots are real 2. The roots are equal if $p=q$ and $r=0$ Which of the above statements is/are correct ?

  1. $1$ only
  2. $2$ only
  3. Both $1$ and $2$
  4. Neither $1$ not $2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation 4(x-p)(x-q) - r^2 = 0 can be written as 4x^2 - 4(p+q)x + 4pq - r^2 = 0. The discriminant D = [4(p+q)]^2 - 4(4)(4pq - r^2) = 16(p^2 + 2pq + q^2) - 16(4pq - r^2) = 16(p^2 - 2pq + q^2 + r^2) = 16[(p-q)^2 + r^2]. Since (p-q)^2 >= 0 and r^2 >= 0, D >= 0, so roots are real. If p=q and r=0, D=0, so roots are equal.

AI explanation

Expanding the equation gives 4x^2 - 4(p+q)x + 4pq - r^2 = 0. The discriminant evaluates to 16(p-q)^2 + 16r^2, which is always non-negative for real numbers, meaning the roots are always real and proving statement 1 is correct. Furthermore, if p equals q and r equals 0, the discriminant becomes 0, meaning the roots are equal and proving statement 2 is correct. Therefore, both 1 and 2 are correct.

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