91Ó°ÊÓ

Understanding the discriminant is crucial when working with quadratic functions. It gives us information about the nature of the roots of a quadratic equation. In the context of inequality problems, particularly where a quadratic function is set to be greater than zero, the discriminant plays an important role. If the quadratic equation represents a function, say f(x) = ax^2 + bx + c, the discriminant \( \Delta \) is expressed as \( \Delta = b^2 - 4ac \).

For the function to have no real roots, indicating that its graph does not intersect the x-axis, the discriminant must be negative (\( \Delta < 0 \)). This ensures the graph of the quadratic function is either entirely above or below the x-axis, depending on the leading coefficient's sign. In our specific problem, because the leading coefficient (\(a = 3\)) is positive, a negative discriminant guarantees that the parabola opens upwards and remains above the x-axis for all real values of \(x\), thus satisfying the original inequality.

Quadratic functions are of the form \( f(x) = ax^2 + bx + c \) where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The graph of this type of function is a parabola. When \(a\) is positive, the parabola opens upwards, and when \(a\) is negative, it opens downwards. Features such as the vertex, axis of symmetry, and direction of opening are calculated using the coefficients of the quadratic function.

In the exercise, the quadratic function given is \(3x^{2}+bx+7\). The positivity of the quadratic coefficient (\(3\)) ensures the parabola opens upwards. We're interested in scenarios where the entire parabola lies above the x-axis, which relates directly to the condition of the inequality being strictly greater than zero (\(f(x)>0\) for all \(x\)). This is a critical aspect for understanding how the values of \(b\) affect the graph and the corresponding inequality condition.

To solve quadratic inequalities like the one in our exercise, we set the quadratic expression within the inequality to be less than zero, as we want the entire expression to be positive for all values of \(x\). The solution to the inequality will give us a range of values for which the inequality holds true.

In our step-by-step solution, after establishing that the discriminant must be negative, we solve the resulting inequality \(b^2 - 84 < 0\) for \(b\). We factored the quadratic expression and found that the value of \(b\) must lie between negative and positive square roots of 84 to maintain the negative discriminant. Simplifying gives us \(|b| < 2 \sqrt{21}\), which translates into a condition that \(b\) must satisfy for the original quadratic inequality to hold true. To solve such inequalities, one may also consider drawing a number line and using test points to determine the intervals that satisfy the inequality, besides algebraic methods.