91Ó°ÊÓ

The roots of a quadratic equation are the solutions to the equation formed by setting the quadratic function equal to zero. For a general quadratic equation in the form of \(ax^2 + bx + c = 0\), the roots \(r_1\) and \(r_2\) can be found using the quadratic formula:

• \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

These roots represent the values of \(x\) where the quadratic function meets the x-axis. For the equation presented in the exercise, \(x^2 + (1-2k)x + (k^2-k-2) = 0\), the roots depend on the value of \(k\). The problem was to find under what conditions the number 3 lies between these roots. This involves assessing whether one root is less than 3 and the other greater, aiding in narrowing down the value \(k\) needs to be.

The discriminant is a key part of the quadratic formula, written as \(D = b^2 - 4ac\). It helps to determine the nature of the roots without actually solving the equation:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, or a repeated root.
• If \(D < 0\), the equation has no real roots, but two complex roots.

In our context, the discriminant was calculated for \(x^2 + (1-2k)x + (k^2-k-2) = 0\). Through simplification, it turned out to be \(D = 9\), indicating the presence of two real and distinct roots. Knowing that both roots are real is crucial as it means it's possible for the number 3 to lie between them as requested in the exercise.

Vieta's formulas provide a way to relate the coefficients of a polynomial to sums and products of its roots. For a quadratic \(ax^2 + bx + c = 0\):

• The sum of the roots \(r_1 + r_2 = -\frac{b}{a}\).
• The product of the roots \(r_1 \cdot r_2 = \frac{c}{a}\).

In the given equation \(x^2 + (1-2k)x + (k^2-k-2) = 0\), Vieta's gives:

• \(r_1 + r_2 = 2k - 1\)
• \(r_1 \cdot r_2 = k^2 - k - 2\)

To ensure 3 is between the roots, inequalities are derived from Vieta's formulas. The conditions worked out showed that 3 can sit between the roots when \(2 < k < 5\). Thus, Vieta's formulas offer a strategic approach to quickly infer relationships and constraints among the roots without solving for them explicitly.