Problem 101 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what values of \(a\) does the function \(2 x^{2}-\left(a^{3}+8 a-1\right) x+a^{2}-4 a\) has the roots of opposite signs?

Step-by-Step Solution

Answer

The short answer is the solution for \(a\) when the discriminant, \(\Delta > 0\).

Step 1: Determine Quadratic Formula Variables

Assign the coefficients from the quadratic equation to the variables a, b, and c. In this case: \(a = 2\), \(b = -(a^{3}+8a-1)\), and \(c = a^{2}-4a\).

Step 2: Compute the discriminant

Use the formula for the discriminant, \(\Delta = b^{2} - 4ac\). Substituting the values of a, b, and c from Step 1 gives us: \(\Delta = [-(a^{3}+8a-1)]^{2} - 4*2*[a^{2} - 4a]\). Simplify this equation to obtain the discriminant, \(\Delta\).

Step 3: Solve for a

The roots of a quadratic function will have opposite signs if and only if the discriminant, \(\Delta\), is greater than 0. Set the equation \(\Delta > 0\) and solve for \(a\).

Key Concepts

Discriminant

In the realm of quadratic equations, the discriminant plays a crucial role in determining the nature of the roots of the equation. For a general quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant, denoted as \(\Delta\), is given by the formula \(\Delta = b^2 - 4ac\).

The discriminant can help us predict the behavior of the solutions:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root, also known as a "repeated" or "double" root.
• If \(\Delta < 0\), the equation has no real roots, but instead two complex conjugate roots.

In our context, to ensure that the quadratic function has roots of opposite signs, it is necessary for the discriminant \(\Delta\) to be greater than 0.

Thus, we start by substituting the given parameters from the problem into the formula for \(\Delta\):
\(\Delta = [-(a^3 + 8a - 1)]^2 - 4 \times 2 \times (a^2 - 4a)\). Simplifying this gives the necessary condition \(\Delta > 0\) for the roots to be of opposite signs.

Opposite Sign Roots

Quadratic equations have interesting properties, one of them being how the sign of the roots affects their product. For the roots to have opposite signs, the product of the roots must be negative. In a quadratic equation \(ax^2 + bx + c = 0\), the product of the roots can be represented by the formula \(c/a\).

If we have \(c = a^2 - 4a\) and \(a = 2\), to determine the conditions for opposite sign roots, we ensure \((a^2 - 4a)/2 < 0\).
This condition implies:

• The value of \(c\) must be negative when divided by the leading coefficient \(a\).
• This ensures that the product of the roots is negative, hence they must have opposite signs.

Thus, solving \((a^2 - 4a)/2 < 0\) along with \(\Delta > 0\) would give us the required values of \(a\) for the quadratic function to satisfy these conditions.

Quadratic Formula

The quadratic formula is the go-to tool for finding the roots of any quadratic equation. It is given by:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
where \(\Delta = b^2 - 4ac\), as mentioned before.

Using the quadratic formula, the two roots are found by evaluating:

• \(x_1 = \frac{-b + \sqrt{\Delta}}{2a}\)
• \(x_2 = \frac{-b - \sqrt{\Delta}}{2a}\)

The discriminant \(\Delta\), being positive, ensures these calculations give two distinct real numbers.

This makes the formula invaluable when finding specific root values, especially determining whether they satisfy conditions such as having opposite signs.
The process involves substituting our known \(b\) and \(c\) values to find roots that match the conditions set forth in the exercise. The roots' signs can then be analyzed, confirming they are opposite if the conditions are right.