TOPIC - QUADRATIC EQUATION(MATHS)

              QUADRATIC EQUATION 

A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The solutions to a quadratic equation, also known as its roots, are the values of 'x' that satisfy the equation. These solutions can be found using several methods, including factoring, completing the square, or the quadratic formula. 

Key Aspects of Quadratic Equations:

Standard Form:

ax² + bx + c = 0, where 'a', 'b', and 'c' are real numbers, and 'a' is not zero. 

Coefficients:

'a' is the quadratic coefficient, 'b' is the linear coefficient, and 'c' is the constant term. 

Roots/Solutions:

The values of 'x' that make the equation true. 

Methods of Solving:

Factoring: Rewriting the quadratic expression as a product of two linear factors. 

Completing the Square: Manipulating the equation to form a perfect square trinomial. 

Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a, provides a direct solution, even if factoring is difficult. 

Discriminant:

The term b² - 4ac within the quadratic formula. Its value determines the nature of the roots:

Positive Discriminant: Two distinct real roots. 

Zero Discriminant: One real root (a repeated root). 

Negative Discriminant: Two complex (non-real) roots. 

Real-life Applications:

Quadratic equations are used in various fields, including physics (projectile motion), engineering, and economics. 

Example:

Consider the equation 2x² + 5x - 3 = 0.

Here, a = 2, b = 5, and c = -3. 

Using the quadratic formula: x = (-5 ± √(5² - 4 * 2 * -3)) / (2 * 2) = (-5 ± √49) / 4 = (-5 ± 7) / 4.

Therefore, the roots are x = 1/2 and x = -3.