Problem 5 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The quadratic in \(t\), such that \(\mathrm{A} \cdot \mathrm{M}\). of its roots in \(A\) and G.M. is \(G\), is (a) \(t^{2}-2 A t+G^{2}=0\) (b) \(t^{2}-2 A t-G^{2}=0\) (c) \(t^{2}+2 A t+G^{2}=0\) (d) none of these

Step-by-Step Solution

Answer

The correct quadratic equation is (a) \( t^2 - 2At + G^2 = 0 \).

Step 1: Understanding the Problem

We are given that the Arithmetic Mean (A) and the Geometric Mean (G) of the roots of a quadratic equation are known. We need to form the corresponding quadratic equation.

Step 2: Recall the Concepts of AM and GM

For a quadratic equation with roots \( \alpha \) and \( \beta \), the Arithmetic Mean of the roots is \( A = \frac{\alpha + \beta}{2} \), and the Geometric Mean is \( G = \sqrt{\alpha \beta} \).

Step 3: Formulate AM Equation

From the AM, we can write \( \alpha + \beta = 2A \).

Step 4: Formulate GM Equation

From the GM, we can express \( \alpha \beta = G^2 \) since \( G = \sqrt{\alpha \beta} \).

Step 5: Write the Standard Quadratic Equation

The standard form for a quadratic equation with roots \( \alpha \) and \( \beta \) is \( t^2 - (\alpha + \beta)t + \alpha \beta = 0 \).

Step 6: Substitute AM and GM into the Equation

By substituting \( \alpha + \beta = 2A \) and \( \alpha \beta = G^2 \), the equation becomes \( t^2 - 2At + G^2 = 0 \).

Step 7: Conclusion

The quadratic equation that fits the given conditions is \( t^2 - 2At + G^2 = 0 \), matching option (a).

Key Concepts

Understanding the Arithmetic Mean

The Arithmetic Mean (AM) is a fundamental concept in mathematics and is often referred to as the "average". This calculation helps us determine what the 'average' value in a data set is. When considering the roots of a quadratic equation, let's denote these roots as \( \alpha \) and \( \beta \). The arithmetic mean gives us an idea of where the middle of these values lies. For two numbers, like our roots, the arithmetic mean is calculated as:

• \( A = \frac{\alpha + \beta}{2} \)

This formula essentially sums the two values and divides by two, giving us the average of our roots. Knowing the arithmetic mean can be particularly helpful when analyzing quadratic equations, as it allows us to easily derive the sum of the roots, which is twice the arithmetic mean.

Grasping the Geometric Mean

The Geometric Mean (GM) is another useful concept, especially when dealing with two numbers that can be related multiplicatively. Unlike the arithmetic mean, which is concerned with the sum of values, the geometric mean considers their product. Given the same roots \( \alpha \) and \( \beta \) of a quadratic equation, the geometric mean is calculated as:

• \( G = \sqrt{\alpha \beta} \)

The geometric mean is particularly useful in understanding the relationship of numbers that scale or grow mathematically. This is because the GM gives you a type of "central" product value for the roots. For quadratic equations, knowing the geometric mean provides us with insight into the product of the roots, which is directly used in formulating the quadratic equation.

Identifying the Roots of a Quadratic Equation

Roots of a quadratic equation are simply the values of \( t \) that satisfy the equation \( at^2 + bt + c = 0 \). In our equation \( t^2 - 2At + G^2 = 0 \), these roots can be represented as \( \alpha \) and \( \beta \). The roots are important because they provide the values that make the quadratic equation true. By using the formulas for arithmetic and geometric means, we have already linked roots to certain numerical expressions. Specifically, the sum of the roots \( \alpha + \beta = 2A \) and the product \( \alpha \beta = G^2 \).To find these roots, we solve the quadratic equation by various methods, like factoring, using the quadratic formula, or completing the square. However, because we use an equation derived from known means, the roots \( \alpha \) and \( \beta \) abide by these relationships inherently due to the design of the equation.

Formulating a Quadratic Equation

Formulating a quadratic equation using its roots is an orderly process consisting of different steps that stem from the relationship between the equation and its roots. We can start forming such an equation using the convolution of these steps:Firstly, recall that any quadratic equation can be expressed as \( t^2 - (\alpha + \beta)t + \alpha \beta = 0 \). In this form, the expression \( \alpha + \beta \) represents the total sum of the roots, while \( \alpha \beta \) represents their product.

• Given \( \alpha + \beta = 2A \), substitute it into the equation for the sum of the roots.
• Given \( \alpha \beta = G^2 \), replace the product in the equation with this expression.

By diligently substituting these derived values back into the equation, we frame it as \( t^2 - 2At + G^2 = 0 \). This structural method ensures that we are correctly formulating a quadratic equation based on known arithmetic and geometric means, leading to the correct mathematical representation, aligned with option (a) from the initial problem setup.