Multiple choice

lf $a$ and $b$ $(\neq 0)$ are the roots of the equation $x^{2}+ax+b=0$, then the least value of $x^{2}+ax +b$ $(x \in R)$ is

  1. $\displaystyle\frac{9}{4}$
  2. $-\displaystyle \frac{9}{4}$
  3. $-\displaystyle \frac{1}{4}$
  4. $ \displaystyle \frac{1}{4}$
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B Correct answer
Explanation

Equation: x^2 + ax + b = 0. Roots are a and b. Sum of roots = a + b = -a, so b = -2a. Product of roots = ab = b. Since b is not 0, a = 1. Then b = -2(1) = -2. The expression is x^2 + x - 2. The minimum value of a quadratic ax^2 + bx + c is at x = -b/(2a) = -1/2. Value = (-1/2)^2 + (-1/2) - 2 = 1/4 - 1/2 - 2 = -2.25 = -9/4.

AI explanation

Since a and b are roots of x^2 + ax + b = 0, substituting them gives a^2 + a^2 + b = 0 and b^2 + ab + b = 0. Solving b(b + a + 1) = 0 with b not equal to 0 yields b = -a - 1, which leads to 2a^2 - a - 1 = 0 and the valid root a = 1, making b = -2. The minimum value of the expression x^2 + x - 2 occurs at x = -1/2, giving (-1/2)^2 + (-1/2) - 2 = -9/4.

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