Multiple choice

If $\alpha, \beta$ are roots of the equation $x^2 + ax + b = 0$ then Find the value of expression $ \dfrac{(\alpha - \beta)^2}{4}$ will be

  1. $\sqrt {\dfrac {a^2 - 4b}{16}}$
  2. $\dfrac{a^2 - 4b}{4}$
  3. $0$

  4. zero
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B Correct answer
Explanation

For x^2 + ax + b = 0, the difference of roots (alpha - beta) is sqrt(D)/a_coeff, where D = a^2 - 4b. Thus, (alpha - beta)^2 = D = a^2 - 4b. The expression (alpha - beta)^2 / 4 is (a^2 - 4b) / 4.

AI explanation

For the quadratic equation x squared plus a x plus b equals 0, the sum of roots is negative a and the product of roots is b. Expanding the square gives alpha minus beta squared as alpha plus beta squared minus four alpha beta, which equals a squared minus four b. Dividing this discriminant expression by four yields a squared minus four b divided by four.

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