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(B) The given quadratic expression is $f(x) = ax^2 + bx + c$ with $a > 0$.To find the minimum value,we complete the square:$f(x) = a(x^2 + \frac{b}{a}

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$\frac{4ac - b^2}{4a}$

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(B) The given quadratic expression is $f(x) = ax^2 + bx + c$ with $a > 0$.To find the minimum value,we complete the square:$f(x) = a(x^2 + \frac{b}{a}x) + c$$f(x) = a(x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 - (\frac{b}{2a})^2) + c$$f(x) = a(x + \frac{b}{2a})^2 - \frac{b^2}{4a} + c$$f(x) = a(x + \frac{b}{2a})^2 + \frac{4ac - b^2}{4a}$Since $a > 0$,the term $a(x + \frac{b}{2a})^2 \ge 0$ for all real $x$.The minimum value occurs when $x = -\frac{b}{2a}$,and the minimum value is $\frac{4ac - b^2}{4a}$.

If $a, b, c$ are real numbers and $a > 0$,what is the minimum value of the quadratic expression $ax^2 + bx + c$ for real $x$?

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