Quadratic Equations

1. Introduction and Historical Background

Quadratic equations arise frequently in real-life situations, such as calculating dimensions of a room given its area and a relationship between length and breadth.
Babylonians were likely the first to solve quadratic equations (finding two numbers with a given sum and product).
Euclid developed a geometrical approach to finding lengths that correspond to solutions of quadratic equations.
Brahmagupta (C.E. 598–665) provided an explicit formula to solve equations of the form ax² + bx = c.
Sridharacharya (C.E. 1025) derived the quadratic formula using the method of completing the square.
Al-Khwarizmi (approx. C.E. 800) and Abraham bar Hiyya Ha-Nasi also contributed significantly to the study of these equations.

A quadratic equation in the variable x is an equation of the form:
ax² + bx + c = 0
where a, b, and c are real numbers and a ≠ 0.
Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation.
Writing the terms in descending order of their degrees yields the standard form of the equation.
Equations may sometimes appear cubic or linear initially but simplify into quadratic equations (e.g., (x + 1)² = 2(x - 3)).

A real number α is called a root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0.
The roots of the quadratic equation are identical to the zeroes of the corresponding quadratic polynomial.
A quadratic equation can have at most two roots.
Method:
- Split the middle term bx into two parts such that their product equals the product of ax² and c.
- Factorise the equation into two linear factors.
- Equate each factor to zero to find the values of x.
Example: To solve 2x² - 5x + 3 = 0, split -5x into -2x and -3x, factorise to get (2x - 3)(x - 1) = 0, yielding roots 3/2 and 1.

The roots of ax² + bx + c = 0 can be calculated using the quadratic formula:
x = -b ± √(b² - 4ac) 2a
The term b² - 4ac is known as the discriminant of the quadratic equation. It determines the nature of the roots.
There are three cases for the nature of roots:
1. Two distinct real roots: If b² - 4ac > 0.
2. Two equal real roots (Coincident roots): If b² - 4ac = 0. (In this case, x = -b/2a).
3. No real roots: If b² - 4ac < 0 (the square root of a negative number is not real).

Quadratic equations are used to solve word problems involving areas, perimeters, and numbers.
When solving real-world problems (like length or time), negative roots are often ignored if they do not make physical sense in the context of the problem.

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