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# quadratic formula

The roots of the quadratic equation

$ax^{2}+bx+c=0\qquad{a,b,c\in\mathbbmss{R},a\neq 0}$ |

are given by the formula

$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$ |

The number $\Delta=b^{2}-4ac$ is called the *discriminant* of the equation.
If $\Delta>0$, there are two different real roots,
if $\Delta=0$ there is a single real root,
and if $\Delta<0$ there are no real roots (but two different complex roots).

Let’s work a few examples.

First, consider $2x^{2}-14x+24=0$. Here $a=2$, $b=-14$, and $c=24$. Substituting in the formula gives us

$x=\frac{14\pm\sqrt{(-14)^{2}-4\cdot 2\cdot 24}}{2\cdot 2}=\frac{14\pm\sqrt{4}}% {4}=\frac{14\pm 2}{4}=\frac{7\pm 1}{2}.$ |

So we have two solutions (depending on whether we take the sign $+$ or $-$): $x=\frac{8}{2}=4$ and $x=\frac{6}{2}=3$.

Now we will solve $x^{2}-x-1=0$. Here $a=1$, $b=-1$, and $c=-1$, so

$x=\frac{1\pm\sqrt{(-1)^{2}-4(1)(-1)}}{2}=\frac{1\pm{\sqrt{5}}}{2},$ |

and the solutions are $x=\frac{1+\sqrt{5}}{2}$ and $x=\frac{1-\sqrt{5}}{2}$.

Keywords:

Algebra, Polynomial

Related:

DerivationOfQuadraticFormula, QuadraticInequality, QuadraticEquationInMathbbC, ConjugatedRootsOfEquation2, QuadraticCongruence

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

12D10*no label found*26A99

*no label found*26A24

*no label found*26A09

*no label found*26A06

*no label found*26-01

*no label found*11-00

*no label found*

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## Quadratic Equation Solver

For anybody who might need a convenient, good-quality calculator, there is an online Quadratic Equation Solver posted here:

http://www.akiti.ca/Quad2Deg.html

It accepts the three coefficients as inputs (a, b, and c) and outputs the two roots, including complex results if applicable.