# Quadratic Equation

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Related to Derivation of quadratic formula: Completing the square

## quadratic equation

[kwä′drad·ik i′kwā·zhən]## Quadratic Equation

an equation of the form *ax*^{2} + *bx* + *c* = 0, where *a, b*, and *c* are any number and are called the coefficients of the equation. A quadratic equation has two roots, which are found by the formulas

The expression *D = b ^{2} − 4ac* is called the discriminant of the quadratic equation. If

*D*> 0, then the roots of the quadratic equation are real and unequal; if

*D*< 0, then the roots areconjugate complex numbers; if

*D*= 0, then the roots are real andequal. The Vièta formulas

*x*

_{1}+

*x*

_{2}= −

*b/a*and x\x

_{2}=

*c/a*link the roots and coefficients of a quadratic equation. The left-hand side of a quadratic equation can be expressed in the form

*α*(

*x*−

*x*

_{2}) (

*x*−

*x*

_{2}). The function

*y*=

*ax*

^{2}+

*bx*+

*c*is called a quadratic trinomial, and its graph is a parabola with the vertex at the point

*M*(−

*b/2a*;

*c*−

*b*

^{2}/4

*a*) and axis of symmetry parallel to the.

*y*-axis; the direction of the branches of the parabola coincides with the sign of

*a*. The solution of the quadratic equation was already known in geometric form to ancient mathematicians.