91Ó°ÊÓ

At the heart of solving any quadratic equation, like the one provided in our exercise, is the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
This formula is derived from the process of completing the square and is a powerful tool that allows us to find the roots of any quadratic equation, which has the general form \(ax^2 + bx + c = 0\). The symbols \(a\), \(b\), and \(c\) represent known values where \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term. 'Discriminant'—the part under the square root sign in the quadratic formula—determines the nature and number of roots of the equation. A positive discriminant results in two distinct real roots, a zero discriminant gives one repeated real root, and a negative discriminant leads to complex roots. In our case, the discriminant is positive, guaranteeing real, albeit opposite, roots.

The sign of the roots in a quadratic equation is an interesting feature that can be determined without explicitly solving the equation. For the given exercise,
\[ x^2 + bx - c = 0 \ where \ b, c > 0 \]
the quadratic formula can be applied to show the roots are of opposite signs. This condition arises due to the structure of the mathematical situation: \(b\) is positive, the constant term \(-c\) is technically negative because we are given that \(c\) is a positive number. The quadratic formula then displays two variations of the root, one involving addition and the other subtraction of the square root term. Given that the denominator is always positive (since it's \(2a\) where \(a = 1\)), the opposing signs come from the numerator. A deeper understanding of the signs can stem from considering the parabola's intersection with the x-axis; if the parabola opens upwards and the \(c\) term is negative, one root must be positive (where the parabola intersects with the axis on the right side), and the other must be negative (the left-side intersection).

Root analysis is an integral part of the study of quadratic equations. It involves understanding the quantitative and qualitative aspects of the roots such as their values, signs, and how they relate to the coefficients of the quadratic equation.
In our exercise, analyzing the roots involved plugging the given constants into the quadratic formula and simplifying. Since the discriminant (\(b^2 + 4c\)) is positive, we knew the roots were real numbers. Furthermore, by examining the numerators of the roots (
\[ x_1 = \frac{{-b + \sqrt{{b^2 + 4c}}}}{{2}} \] and
\[ x_2 = \frac{{-b - \sqrt{{b^2 + 4c}}}}{{2}} \]),
we could conclude that they had to have opposite signs, as the addition and subtraction of the same positive square root value would result in numerically different outcomes. The denominator, being positive, does not alter the analysis of the root signs. This facet of the root analysis is crucial because it can predict the behavior of a quadratic function without precisely pinpointing the roots' exact values.