Problem 69 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the values of \(a\) for which \(\left(a^{2}-1\right) x^{2}+2(a-1) x+2\) is positive for any \(x\).

Step-by-Step Solution

Answer

The trinomial is always positive for \(a < -1\) or \(a > 1\), but not for \(a = 1\).

Step 1: Identify the coefficients

We identify the coefficients in the trinomial equation \(ax^{2}+bx+c\), which are \(a=a^{2}-1\), \(b=2(a-1)\) and \(c=2\) respectively.

Step 2: the condition for real roots

The condition for the quadratic equation to have real roots is that the discriminant \(b^{2}-4ac ≥ 0\) must be satisfied. However, we want the discriminant to be negative, as stated in the analysis. So, the condition for the quadratic equation not to have real roots (thus being always positive or always negative) is that \(b^{2}-4ac < 0\). Let’s express this inequality in terms of \(a\): \(b^2 - 4ac = [2(a - 1)]^2 - 4 * (a^2 - 1) * 2 < 0\).

Step 3: Simplify the inequality

After carrying out the multiplication and subtraction operations, the inequality simplifies to \(4a^2 - 12a + 4 - 8a^2 + 8 < 0\). Further simplification results in \(-4a^2 +12a - 4 < 0\).

Step 4: Solve the inequality

To solve the inequality \(-4a^2 +12a - 4 < 0\), we could factor out a \(-4\) and rearmanget the terms to obtain \(4(1-a)^2 ≥ 0\). Since \(4(1-a)^2\) is a square (and any square is greater or equal than zero), it follows from our inequality that it has to be strictly greater than zero. This is only possible if \(1-a ≠ 0\), which implies that \(a ≠ 1\).

Step 5: Check that \(a^2 - 1 > 0\)

We know from the analysis that the coefficient of \(x^2\) (\(a^2 - 1\)) must be positive in order for the trinomial to always be positive. By solving the simple inequality \(a^2 - 1 > 0\), we find that \(a < -1\) or \(a > 1\). So, putting all together, in order for the trinomial to always be positive, the solutions must satisfy both \(a ≠ 1\) and \(a < -1\) or \(a > 1\).

Key Concepts

Discriminant

The discriminant is a key factor in determining the nature of the roots of a quadratic equation. Its formula is given by the expression \(b^2 - 4ac\), derived from the general quadratic equation \(ax^2 + bx + c = 0\). Depending on the value of the discriminant:

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the roots are real and repeated, meaning there is exactly one real root.
• If \(b^2 - 4ac < 0\), the equation has no real roots, meaning the parabola does not intersect the x-axis.

In the exercise, we aim for the expression to be always positive, requiring the discriminant to be negative (\(b^2 - 4ac < 0\)). This ensures no real roots exist, keeping the solution above the x-axis. Simplifying and solving this expression for given coefficients helps identify the conditions under which our quadratic is always positive.

Solving Inequalities

Solving inequalities is a crucial part of understanding the behavior of quadratic expressions. Whereas equations give specific solutions, inequalities describe a range of solutions. In the exercise, we encounter the inequality \(-4a^2 + 12a - 4 < 0\). To solve it, we aim to find values of \(a\) that keep the expression negative.Steps to solve the inequality:

• Identify expressions that can be simplified or factored.
• Rearrange terms if necessary and factor out common factors.
• Analyze any squares or factored terms to understand how they affect the inequality.

We factored and rearranged to obtain \(4(1-a)^2 \ge 0\). Since squares of real numbers are always non-negative, the solution only holds if \(a eq 1\). Thus, our conditions for \(a\) include \(a < -1\) or \(a > 1\), avoiding \(a = 1\) to ensure the expression is always positive.

Quadratic Coefficients

Quadratic coefficients, namely \(a\), \(b\), and \(c\), are vital in defining the shape and position of a parabola described by a quadratic equation. In the exercise equation \(\left(a^2-1\right)x^2 + 2(a-1)x + 2\), each term involves these coefficients. Understanding their role helps in manipulating and solving the expression.The coefficients determine:

• \(a\): affects the width and direction of the parabola (upward if \(a > 0\), and downward if \(a < 0\)).
• \(b\): affects the tilt and axis of symmetry of the parabola.
• \(c\): determines the y-intercept, the point where the parabola crosses the y-axis.

In our case, ensuring \(a^2 - 1 > 0\) maintains that the parabola opens either entirely above or below the x-axis.This contributes crucially to solving the inequality and conditions for \(a\) that allow the quadratic expression to be strictly positive for all \(x\). Thus, with \(a < -1\) or \(a > 1\), the expression stays positive, ensuring specific parabola placements regarding the x-axis.