Problem 111 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(a, b, c\) are real numbers, \(a \neq 0\). If \(\alpha\) is a root of \(a^{2} x^{2}+b x+c=0, \beta\) is a root of \(a^{2} x^{2}-b x-c=0\) and \(0<\alpha<\beta\), then show that the equation \(a^{2} x^{2}+2 b x+2 c=0\) has a root \(\gamma\) that always lies between \(\alpha\) and \(\beta\).

Step-by-Step Solution

Answer

In conclusion, we have proven that there exists a root, \(\gamma\), of the equation \(a^2x^2 + 2bx + 2c = 0\) that always lies between \(\alpha\) and \(\beta\). We did this by analyzing the given equations, finding a real root of the third equation, and using the Intermediate Value Theorem.

Step 1: Analyze the given equations

We have two quadratic equations: 1. \(a^2x^2 + bx + c = 0\), with root \(\alpha\), and 2. \(a^2x^2 - bx - c = 0\), with root \(\beta\). We have to find if there exists a root, \(\gamma\), that lies between \(\alpha\) and \(\beta\) for the equation: \(a^2x^2 + 2bx + 2c = 0\).

Step 2: Find a real root of the third equation

Let's denote the third equation as: \(f(x) = a^2x^2 + 2bx + 2c\). We know that: \(f(\alpha) = a^2\alpha^2 + 2b\alpha + 2c\), and \(f(\beta) = a^2\beta^2 + 2b\beta + 2c\). Since \(\alpha\) and \(\beta\) are roots of the first and second equations respectively, we can write: \(f(\alpha) = 2(a^2\alpha^2 + b\alpha + c) = 2(0) = 0\), and \(f(\beta) = 2(a^2\beta^2 - b\beta - c) = 2(0) = 0\).

Step 3: Use the Intermediate Value Theorem

Now we need to prove that there exists a root, \(\gamma\), between \(\alpha\) and \(\beta\). We know that: \(0 < \alpha < \beta\) Let's consider the function: \(g(x) = f(x) - f(\alpha) - f(\beta) = a^2x^2 + 2bx + 2c - 2(a^2\alpha^2 + b\alpha + c) - 2(a^2\beta^2 - b\beta - c)\) For \(\alpha\) and \(\beta\), we have: \(g(\alpha) = 0\), and \(g(\beta) = 0\). Since \(g(x)\) is a continuous function, and \(0 < \alpha < \beta\), we can apply the Intermediate Value Theorem. There exists at least one root, \(\gamma\), that lies between \(\alpha\) and \(\beta\) for the function \(g(x)\), such that: \(g(\gamma) = f(\gamma) - f(\alpha) - f(\beta) = 0\). Thus, \(f(\gamma) = f(\alpha) + f(\beta)\) has a root between \(\alpha\) and \(\beta\).

Step 4: Conclusion

In conclusion, we have proven that there exists a root, \(\gamma\), of the third equation \(a^2x^2 + 2bx + 2c = 0\) that always lies between \(\alpha\) and \(\beta\).

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is an essential concept in calculus, especially when it comes to understanding the behavior of functions. It states that for any continuous function that provides an output for every input within a certain interval, if the function takes on any two values at either end of the interval, then it must take any value in between these two values at some point within the interval. This theorem is fundamental for proving the existence of roots in equations.

For example, in the quadratic equation problem we're examining, we are given a continuous function that changes from one value to another as it moves from \(\alpha\) to \(\beta\). By the IVT, if the function's value at \(\alpha\) and \(\beta\) includes zero (which is between any positive and negative number), there must be at least one point between \(\alpha\) and \(\beta\) where the function crosses the x-axis — this point is our root, \(\gamma\). In simple terms, if you're traveling from one elevation to another, you have to pass through every elevation in between, which is what the IVT guarantees for continuous functions.

Quadratic Equation Analysis

Quadratic equation analysis involves examining the nature of the roots of a quadratic function and how they relate to the function's graph. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \) where \( a, b, \) and \( c \) are constants, and \( a \) is non-zero. The roots, or solutions, are the values of \( x \) where the graph of the quadratic function intersects the x-axis.

By analyzing the given problem, we recognize that the roots \(\alpha\) and \(\beta\) are critical to finding the root \(\gamma\). Unlike simple quadratic equations, the equations here are transformed, but their roots are still governed by the same principles. Finding \(\gamma\) that lies between \(\alpha\) and \(\beta\) involves understanding not just the individual roots but also how changes in the coefficients affect the position and nature of these roots within a certain interval. This analysis usually combines algebraic methods with the graphical understanding of the quadratic function.

Continuous Functions in Calculus

Continuous functions in calculus are functions that do not have any gaps, jumps, or breaks in their graphs. They are the cornerstone of many theorems and principles in calculus, including the IVT. For a function to be considered continuous at a point, the function value (limit) must approach the same number from both the left and right side as it does at the point itself. Moreover, for a function to be continuous on an interval, it must be continuous at every point in that interval.

In our quadratic equation context, the function \( f(x) = a^2x^2 + 2bx + 2c \) is a polynomial function, and all polynomials are continuous everywhere. This is why we can confidently apply the IVT to prove that a root \(\gamma\) exists between \(\alpha\) and \(\beta\). Understanding continuity is vital because it assures us that the function behaves predictably, and there are no unexpected changes in its output as long as the input changes smoothly within the given interval.