Question

If \(a\) and \(b\) are non-zero roots of \(6x^2+ax+b=0\), then the least value of \(x^2+ax+b\) is

• A. \(\dfrac23\)
• B. \(-\dfrac94\)
• C. \(\dfrac94\)
• D. \(1\)

Hint:

Minimum of \(px^2+qx+r\) occurs at \(x=-\frac q{2p}\).

Answer 1

The Correct Option is B

Step 1: Since roots are \(a,b\), \[ a+b=-\frac{a}{6},\quad ab=\frac{b}{6}. \]
Step 2: The quadratic \(x^2+ax+b\) has minimum at \[ x=-\frac a2. \]
Step 3: \[ \text{Minimum value}=b-\frac{a^2}{4}=-\frac94. \]