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Quadratic equation

Author: spiyr

Quadratic equationExample discriminant signs
■ <0: x²+¹⁄₂
■ =0: −⁴⁄₃x²+⁴⁄₃x−¹⁄₃
■ >0: ³⁄₂x²+¹⁄₂x−⁴⁄₃

A quadratic equation with real or complex coefficients has two solutions (or roots), not necessarily distinct, which may or may not be real, given by the quadratic formula:

Discriminant

In the above formula, the expression underneath the square root sign

is called the discriminant of the quadratic equation.

A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

If the discriminant is positive, there are two distinct roots, both of which are real numbers.
For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers-in other cases they may be quadratic irrationals.
- If the discriminant is zero, there is exactly one distinct real root, sometimes called a double root:
- If the discriminant is negative, there are no real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other:
- where, $\begin{align}|D|\end{align}$ is the absolute value(+ve) and $\begin{align}i \end{align}$ is the $\begin{align}{\sqrt {-1}}\end{align}$
- Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
- Example discriminant signs
  ■ <0: x²+¹⁄₂
  ■ =0: −⁴⁄₃x²+⁴⁄₃x−¹⁄₃
  ■ >0: ³⁄₂x²+¹⁄₂x−⁴⁄₃