If the roots of the equation x^2-p x+q=0 differ by unity, then prove that p^2=4 q+1.

Name: Problem 39, Chapter 8: Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry
Uploaded: 2021-09-05T16:12:29-08:00
Duration: 3 min 47 s
Channel: Talha Khalid
Description: If the roots of the equation x^2-p x+q=0 differ by unity, then prove that p^2=4 q+1.

step 1 In this question, we have the condition that the roots of the equation differ by unity or we can

If the roots of the equation x^2-p x+q=0 differ by unity,...

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in one variable with the standard form ax² + bx + c = 0. Its solutions, or roots, can be found using methods such as factoring, completing the square, or the quadratic formula. Understanding quadratic equations is fundamental since many problems in algebra involve finding the relationship between the coefficients and the roots.

Vieta's Formulas

Vieta's formulas relate the sum and product of the roots of a quadratic equation to its coefficients. For a quadratic equation of the form x² - p*x + q = 0, the sum of the roots is equal to p and the product of the roots is equal to q. These relationships are crucial for establishing connections between the algebraic properties of the roots and the coefficients of the polynomial.

Root Difference Identity

The root difference identity stems from the relation between the sum and product of the roots and the difference between them. Specifically, for any two numbers, (? - ?)² can be expressed as (? + ?)² - 4??. Applying this identity to the roots of a quadratic equation provides a method to relate the square of the difference of the roots directly to the coefficients of the equation. This identity is particularly useful when given a specific numerical difference between the roots, as it leads to algebraic conditions on the coefficients.

Transcript

Please give Ace some feedback