Roots of quadratic equations

A quadratic equation is a second order polynomial with three coefficients a, b, c. The general form is axe^2 + bx + c = 0, with a ≠ 0.
The roots of a quadratic equation are the values of x that satisfy the equation.
The solutions can be real or complex. If the discriminant (b^2 - 4ac) is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and identical. If the discriminant is negative, the roots are complex and conjugate pairs.
The quadratic formula, x = [ -b ± sqrt(b^2 - 4ac) ] / 2a, can be used to find the roots of any quadratic equation.
When a quadratic equation is set to zero, the roots can also represent the x-intercepts of the corresponding quadratic graph.
For a quadratic equation, the sum of the roots is given by -b/a and the product of the roots is given by c/a.
Completing the square is another method to solve quadratic equations, especially useful when the coefficient of x^2 is not 1.
The discriminant can also describe the nature of the relations between roots and coefficients in a quadratic equation. The sign of the discriminant tells whether the roots are rational or irrational.
Quadratic equations may be rearranged and rewritten using algebraic methods to make solving them simpler or to highlight different aspects of their solutions.
Quadratic equations and their roots have important applications in many areas of pure mathematics, including area and volume calculation, optimisation problems, and algebraic proof.