Quadratic Equations

 Quadratic Equations

 

Quadratic Equations

A quadratic equation in the variable `x` is an equation of the form `ax^2 + bx + c = 0`, where a, b, c are real numbers, a `ne` 0. 

For example, `2x^2 + x – 300 = 0` is a quadratic equation. Similarly, `2x^2 – 3x + 1 = 0`, `4x – 3x^2 + 2 = 0` and `1 – x^2 + 300 = 0` are also quadratic equations.

The standard form of a quadratic equation

Any equation of the form `p(x) = 0`, where `p(x)` is a polynomial of degree 2, is a quadratic equation. But when we write the terms of `p(x)` in descending order of their degrees, then we get the standard form of the equation. That is, `ax^2 + bx + c = 0, a ne 0` is called the standard form of a quadratic equation.

Quadratic formula

The roots of a quadratic equation `ax^2 + bx + c = 0` are given by

`frac{-b ± sqrt (b^2 - 4ac)}{2a}`, provided `b^2 – 4ac ≥ 0`.

Nature of Roots 

A quadratic equation `ax^2 + bx + c = 0` has

(i) two distinct real roots, if `b^2 – 4ac > 0`,

(ii) two equal roots (i.e., coincident roots), if `b^2 – 4ac = 0`, and

(iii) no real roots, if `b^2 – 4ac < 0`.