Problem 107 - A Complete Resource Book in Mathematics for JEE Main

Question

The equation \(a x^{4}-2 x^{2}-(a-1)=0\) will have real and unequal roots if (A) \(a<0, a \neq 1\) (B) \(a>0, a \neq 1\) (C) \(0

Step-by-Step Solution

Answer

Option (D): None of these.

Step 1: Understand the Problem

We need to analyze the given quartic equation for real and unequal roots. The equation is \( ax^4 - 2x^2 - (a-1) = 0 \). This can be considered as a quadratic in terms of \( x^2 \), i.e., \( ax^2)^2 - 2x^2 - (a-1) = 0 \).

Step 2: Substitute and Simplify

Let \( y = x^2 \). Then the equation becomes \( ay^2 - 2y - (a-1) = 0 \). This is a quadratic equation in \( y \).

Step 3: Apply Discriminant Condition for Quadratic

For a quadratic equation \( ay^2 + by + c = 0 \) to have real and unequal roots, the discriminant should be greater than zero. The discriminant \( \Delta \) is given by \( b^2 - 4ac \).

Step 4: Calculate the Discriminant

Here, \( a = a \), \( b = -2 \), and \( c = -(a-1) \). The discriminant is \((-2)^2 - 4 \times a \times -(a-1) = 4 + 4a(a-1) = 4 + 4a^2 - 4a \).

Step 5: Set the Discriminant Greater than Zero

For real and unequal roots, \( \Delta = 4 + 4a^2 - 4a > 0 \). Simplifying gives \( 4a^2 - 4a + 4 > 0 \).

Step 6: Analyze the Quadratic in 'a'

Divide the inequality by 4: \( a^2 - a + 1 > 0 \). This quadratic does not factor neatly, so check its discriminant: \( (-1)^2 - 4\times 1\times 1 = 1 - 4 = -3 \), which is negative.

Step 7: Determine Conclusion from Discriminant Sign

Since the quadratic \( a^2 - a + 1 \) has a negative discriminant, it is always positive for all real \( a \). Hence, there is no restriction on \( a \) for real and unequal roots. So, option (D) could be correct.

Key Concepts

Discriminant

The discriminant is a key part of determining the nature of the roots of a quadratic equation. It is essentially a value calculated from the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). The formula for the discriminant \( \Delta \) is given by:

• \( \Delta = b^2 - 4ac \)

This simple calculation tells us a lot about the roots without actually solving the equation.
Each potential outcome guides us:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (a repeated root).
• If \( \Delta < 0 \), the roots are complex, which means no real roots exist.

Understanding the discriminant allows you to predict the types of roots you can expect from the equation.

Real and Unequal Roots

For an equation to have real and unequal roots, its discriminant must be greater than zero. In simple terms, the requirement \( \Delta > 0 \) ensures that the roots will be real numbers and different from each other. This condition is significant in algebra because equal roots suggest repeated solutions, showing the equation's symmetry or specific behaviors in graphs.
In the context of \( ay^2 - 2y - (a-1) = 0 \), we want the discriminant \( \Delta = 4a^2 - 4a + 4 \) to be more than zero to ensure the roots are real and distinct. Achieving real and differing roots often indicates that the solutions will intersect the x-axis at two separate points if graphed.
Therefore, establishing \( a \) such that the discriminant remains positive is crucial for solving the problem correctly.

Quartic Equation

A quartic equation is a polynomial equation of degree four, meaning the highest power of the variable is four. Quartic equations can be quite complex, involving several roots potentially. The generic form is:

• \( ax^4 + bx^3 + cx^2 + dx + e = 0 \)

These equations can have up to four real roots, combinations of real and complex roots, or two pairs of complex conjugates.
The given problem, \( ax^4 - 2x^2 - (a-1) = 0 \), can appear daunting. But, by treating it like a quadratic equation in terms of \( y = x^2 \), it simplifies the process substantially. This method reduces the complexity and helps pinpoint whether the equation has real roots without delving into tricky algebraic manipulations.

Substitution Method

The substitution method is a powerful algebraic technique used to simplify complex problems. By substituting one variable for another expression, we can transform difficult equations into more manageable forms. For instance, in our problem statement:

• Let \( y = x^2 \) to convert the quartic equation \( ax^4 - 2x^2 - (a-1) = 0 \) into the quadratic form \( ay^2 - 2y - (a-1) = 0 \).

This substitution allows you to apply the quadratic formula or inspect the discriminant to assess the nature of the roots more straightforwardly.
Using substitutions can streamline calculations and analyses, especially when dealing with high-degree polynomials or when specific values, like roots, need to be determined. It’s an essential skill in both high school algebra and higher mathematics, simplifying your work and often revealing insights into the problem's structure.