Quadratic Equations

Quadratic equations are a type of polynomial equation of the second degree, fundamental in algebra.

Definition

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where:

'x' represents an unknown variable,
'a', 'b', and 'c' are constants, with 'a' ≠ 0. 

The numbers a, b, and c are called the coefficients of the equation. The term a is the coefficient of x^2 (the quadratic term), b is the coefficient of x (the linear term), and c is the constant term.

Properties

The solutions to a quadratic equation are given by the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a). The term inside the square root, b^2 - 4ac, is called the discriminant.

The discriminant helps determine the nature of the roots of the quadratic equation:

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has one real root (or a repeated root).
• If the discriminant is negative, the equation has two complex roots.

When graphed, a quadratic equation forms a curve called a parabola. The vertex of the parabola is the maximum or minimum point of the curve, depending on the sign of the coefficient a.

Applications

Quadratic equations are widely used in many fields, including physics, engineering, and finance. They can model various real-world situations, such as projectile motion, the shape of satellite dishes and bridges, and profit maximization in business.