Suppose that the equation f(x) = x^{2} + bx + | vedclass

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(A) Given that $\alpha$ and $\beta$ are the roots of $f(x) = x^{2} + bx + c = 0$.By the properties of quadratic equations, the sum of roots is $\alpha

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(A) Given that $\alpha$ and $\beta$ are the roots of $f(x) = x^{2} + bx + c = 0$.By the properties of quadratic equations, the sum of roots is $\alpha + \beta = -b$.The vertex of the parabola $y = f(x)$ occurs at $x = -\frac{b}{2} = \frac{\alpha + \beta}{2}$.Now, find the derivative of $f(x)$:$f'(x) = \frac{dy}{dx} = 2x + b$.At the point $x = \frac{\alpha + \beta}{2} = -\frac{b}{2}$, the slope of the tangent is:$f'\left(-\frac{b}{2}\right) = 2\left(-\frac{b}{2}\right) + b = -b + b = 0$.$A$ slope of $0$ indicates that the tangent line is horizontal, meaning it is parallel to the $x$-axis.Therefore, the angle between the tangent and the positive direction of the $x$-axis is $0^{\circ}$.

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