quadratic formula

The roots of the quadratic equation

$$a{x}^2+bx+c=0\mathit{\hspace{1em}\hspace{1em}}a,b,c\in ℝ,a\ne 0$$

are given by the formula

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

The number $\mathrm{\Delta }=b^2-4ac$ is called the discriminant of the equation. If $\mathrm{\Delta }>0$, there are two different real roots, if $\mathrm{\Delta }=0$ there is a single real root, and if 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 $x=\frac{1-\sqrt{5}}{2}$.

Title	quadratic formula
Canonical name	QuadraticFormula
Date of creation	2013-03-22 11:46:15
Last modified on	2013-03-22 11:46:15
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	13
Author	yark (2760)
Entry type	Theorem
Classification	msc 12D10
Classification	msc 26A99
Classification	msc 26A24
Classification	msc 26A09
Classification	msc 26A06
Classification	msc 26-01
Classification	msc 11-00
Related topic	DerivationOfQuadraticFormula
Related topic	QuadraticInequality
Related topic	QuadraticEquationInMathbbC
Related topic	ConjugatedRootsOfEquation2
Related topic	QuadraticCongruence