Editing Elementary algebra (section)

=== Quadratic equations ===
{{Main|Quadratic equation}}
[[File:Quadratic-equation.svg|thumb|right|Quadratic equation plot of <math>y = x^2 + 3x - 10</math> showing its roots at <math>x = -5</math> and <math>x = 2</math>, and that the quadratic can be rewritten as <math>y = (x + 5)(x - 2)</math>
]]
A quadratic equation is one which includes a term with an exponent of 2, for example, <math>x^2</math>,<ref>Mary Jane Sterling, ''Algebra II For Dummies'', Publisher: John Wiley & Sons, 2006, {{ISBN|0471775819}}, 9780471775812, 384 pages, [https://books.google.com/books?id=_0rTMuSpTY0C&dq=quadratic+equations&pg=PA37 page 37]</ref> and no term with higher exponent. The name derives from the Latin ''quadrus'', meaning square.<ref>John T. Irwin, ''The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story'', Publisher JHU Press, 1996, {{ISBN|0801854660}}, 9780801854668, 512 pages, [https://books.google.com/books?id=jsxTenuOQKgC&dq=quadratic+quadrus&pg=PA372 page 372]</ref> In general, a quadratic equation can be expressed in the form <math>ax^2 + bx + c = 0</math>,<ref>Sharma/khattar, ''The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E'', Publisher Pearson Education India, 2010, {{ISBN|8131723631}}, 9788131723630, 1248 pages, [https://books.google.com/books?id=2v-f9x7-FlsC&dq=quadratic%20equations%20%20ax2%20%2B%20bx%20%2B%20c%20%3D%200&pg=RA13-PA33 page 621]</ref> where {{mvar|a}} is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term <math>ax^2</math>, which is known as the quadratic term. Hence <math>a \neq 0</math>, and so we may divide by {{mvar|a}} and rearrange the equation into the standard form

: <math>x^2 + px + q = 0 </math>

where <math>p = \frac{b}{a}</math> and <math>q = \frac{c}{a}</math>. Solving this, by a process known as [[completing the square]], leads to the [[quadratic formula]]

:<math>x=\frac{-b \pm \sqrt {b^2-4ac}}{2a},</math>

where [[plus–minus sign|the symbol "±"]] indicates that both

: <math> x=\frac{-b + \sqrt {b^2-4ac}}{2a}\quad\text{and}\quad x=\frac{-b - \sqrt {b^2-4ac}}{2a}</math>

are solutions of the quadratic equation.

Quadratic equations can also be solved using [[factorization]] (the reverse process of which is [[polynomial expansion|expansion]], but for two [[linear function|linear terms]] is sometimes denoted [[FOIL rule|foiling]]).  As an example of factoring:

: <math>x^{2} + 3x - 10 = 0, </math>

which is the same thing as

: <math>(x + 5)(x - 2) = 0. </math>

It follows from the [[zero-product property]] that either <math>x = 2</math> or <math>x = -5</math> are the solutions, since precisely one of the factors must be equal to [[zero]]. All quadratic equations will have two solutions in the [[complex number]] system, but need not have any in the [[real number]] system. For example,

: <math>x^{2} + 1 = 0 </math>

has no real number solution since no real number squared equals −1.
Sometimes a quadratic equation has a root of [[multiplicity (mathematics)|multiplicity]] 2, such as:

: <math>(x + 1)^2 = 0. </math>

For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as

:<math>[x-(-1)][x-(-1)]=0.</math>

====Complex numbers====

All quadratic equations have exactly two solutions in [[complex numbers]] (but they may be equal to each other), a category that includes [[real number]]s, [[imaginary number]]s, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation

:<math>x^2+x+1=0</math>

has solutions

:<math>x=\frac{-1 + \sqrt{-3}}{2}  \quad \quad \text{and} \quad \quad x=\frac{-1-\sqrt{-3}}{2}.</math>

Since <math>\sqrt{-3}</math> is not any real number, both of these solutions for ''x'' are complex numbers.