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quadratic formula (Theorem)

The roots of the quadratic equation

\begin{displaymath} ax^2+bx+c=0\qquad{a,b,c\in\mathbbmss{R},a\neq 0} \end{displaymath}

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 $$ x=\frac{14\pm \sqrt{(-14)^2-4\cdot2\cdot24}}{2\cdot 2} =\frac{14\pm\sqrt{4}}{4} =\frac{14\pm2}{4} =\frac{7\pm1}{2}. $$ 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 $x=\frac{1-\sqrt{5}}{2}$ .




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See Also: derivation of quadratic formula, quadratic inequality, quadratic equation in $\mathbb{C}$, conjugated roots of equation, quadratic congruence

Keywords:  Algebra, Polynomial

Attachments:
derivation of quadratic formula (Proof) by mathcam
properties of quadratic equation (Result) by pahio
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Cross-references: solutions, complex roots, real, equation, discriminant, number, formula, quadratic equation, roots
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This is version 8 of quadratic formula, born on 2001-10-15, modified 2007-11-02.
Object id is 227, canonical name is QuadraticFormula.
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Classification:
AMS MSC12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )

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Quadratic Equation Solver by DavidB52 on 2009-10-18 21:30:50
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.
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