Problem 32 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If the sum of the roots of \(a x^{2}+b x+c=0\) be equal to sum of their squares, prove that \(2 a c=a b+b^{2}\).

Step-by-Step Solution

Answer

Using the relationships of roots and coefficients of a quadratic equation and a little bit of algebraic manipulation, we have proven that for a quadratic equation \(ax^2 + bx + c = 0\), if the sum of its roots is equal to the sum of the squares of the roots, then 2ac = ab + b^2.

Step 1: Sum and Product of Roots

Let's denote the roots of the equation \(ax^2 + bx + c = 0\) as α and β. So, by the relationships of roots and coefficients of a quadratic equation, we have (1) α + β = -b/a, which is the sum of the roots, and (2) αβ = c/a, which is the product of the roots.

Step 2: Equating given Condition to the Formula

It's given that the sum of the roots is equal to the sum of the squares of the roots. So, α + β = α^2 + β^2. But we know that (α + β)^2 = α^2 + β^2 + 2αβ. So, let's equate these two equations and solve to find the value of αβ.

Step 3: Calculating and Substituting Value of αβ

From the above equation we get, α^2 + β^2 = α + β − 2αβ. We already know that α + β = -b/a and αβ = c/a. Let's substitute these in our equation. So, we get (-b/a)^2 = -b/a - 2c/a.

Step 4: Final Step: Proving the Required Equations

By simplifying the above equation and transferring terms, we get 2ac = ab + b^2 which is the required equation to prove.

Key Concepts

Sum and Product of Roots

When dealing with quadratic equations of the form \(ax^2 + bx + c = 0\), understanding the sum and product of the roots is crucial. The roots of this equation can be denoted as \(\alpha\) and \(\beta\). Using the relationships of roots and coefficients, we find:

• The sum of the roots \(\alpha + \beta\) is given by \(-\frac{b}{a}\).
• The product of the roots \(\alpha \beta\) is given by \(\frac{c}{a}\).

These formulas arise from the factorization of quadratic equations. They are fundamental because they correlate the roots \(\alpha\) and \(\beta\) directly to the coefficients \(b\) and \(c\) of the equation. This allows us to explore deeper properties of quadratic equations and use these relationships for further analysis.

Relationship of Roots and Coefficients

The relationship of roots \(\alpha\) and \(\beta\) with the coefficients \(-\frac{b}{a}\) and \(\frac{c}{a}\) is pivotal for solving equations efficiently without knowing the explicit values of \(\alpha\) and \(\beta\). To comprehend how this relationship is applied:

• The expression \((\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha\beta\) gives us a tool to expand expressions involving roots.
• By setting \(\alpha + \beta = \alpha^2 + \beta^2\), implied in the problem, we leverage the root-coefficient relationships to establish new connections.

By substituting the values from our known identities, we transform and simplify complex root expressions, linking back our findings to original coefficients of the quadratic, displaying a profound symmetry in algebra.

Proof in Algebra

Proving results like \(2ac = ab + b^2\) involves a structured approach using algebraic identities to validate given conditions. In this exercise, the goal is to demonstrate an equality starting from the derived conditions:

• We began by recognizing the condition \(\alpha + \beta = \alpha^2 + \beta^2\). Using the identity \((\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha\beta\), we substituted and reorganized terms.
• By knowing \((\alpha + \beta)^2 = \frac{b^2}{a^2}\) and simplifying, we equated it to \(-\frac{b}{a} - 2\frac{c}{a}\).
• After further simplification, we achieved the identity \(2ac = ab + b^2\), thus confirming the equation provided.

Proofs in algebra demand careful manipulation of identities and conditions, ensuring that each step logically leads to the desired conclusion. This enhances problem-solving skills and fosters a deeper understanding of algebraic relationships.