Editing Quadratic equation (section)

{{Short description|Polynomial equation of degree two}}

In [[mathematics]], a '''quadratic equation''' ({{etymology|la|{{wikt-lang|la|quadratus}}|[[square (algebra)|square]]}}) is an [[equation]] that can be rearranged in standard form as<ref>{{cite book |title=Intermediate Algebra with Trigonometry |author1=Charles P. McKeague |edition=reprinted |publisher=Academic Press |year=2014 |isbn=978-1-4832-1875-5 |page=219 |url=https://books.google.com/books?id=e4_iBQAAQBAJ}} [https://books.google.com/books?id=e4_iBQAAQBAJ&pg=PA219 Extract of page 219]</ref>
<math display=block>ax^2 + bx + c = 0\,,</math>
where the [[variable (mathematics)|variable]] {{math|''x''}} represents an unknown number, and {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} represent known numbers, where {{math|''a'' ≠ 0}}. (If {{math|''a'' {{=}} 0}} and {{math|''b'' ≠ 0}} then the equation is [[linear equation|linear]], not quadratic.) The numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are the ''[[coefficient]]s'' of the equation and may be distinguished by respectively calling them, the ''quadratic coefficient'', the ''linear coefficient'' and the ''constant coefficient'' or ''free term''.<ref>Protters & Morrey: "Calculus and Analytic Geometry. First Course".</ref>

The values of {{mvar|x}} that satisfy the equation are called ''[[solution (mathematics)|solutions]]'' of the equation, and ''[[zero of a function|roots]]'' or ''[[zero of a function|zeros]]'' of the [[quadratic function]] on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a [[double root]]. If all the coefficients are [[real number]]s, there are either two real solutions, or a single real double root, or two [[complex number|complex]] solutions that are [[complex conjugate]]s of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be [[Factorization|factored]] into an equivalent equation<ref>{{cite book |title=Princeton Review SAT Prep, 2021: 5 Practice Tests + Review & Techniques + Online Tools |author1=The Princeton Review |edition= |publisher=Random House Children's Books |year=2020 |isbn=978-0-525-56974-9 |page=360 |url=https://books.google.com/books?id=IrrQDwAAQBAJ}} [https://books.google.com/books?id=IrrQDwAAQBAJ&pg=PA360 Extract of page 360]</ref> 
<math display=block>ax^2+bx+c=a(x-r)(x-s)=0</math>
where {{Mvar|r}} and {{Mvar|s}} are the solutions for {{Mvar|x}}. 

The [[quadratic formula]]
<math display=block>x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}</math>
expresses the solutions in terms of {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}. [[Completing the square]] is one of several ways for deriving the formula.

Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.<ref>{{cite book |title=Indra's Pearls: The Vision of Felix Klein |author1=David Mumford |author2=Caroline Series |author3=David Wright |edition=illustrated, reprinted |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-35253-6 |page=37 |url=https://books.google.com/books?id=XFE3jmSEfC8C}} [https://books.google.com/books?id=XFE3jmSEfC8C&pg=PA37 Extract of page 37]</ref><ref>{{cite book |title=Mathematics in Action Teachers' Resource Book 4b |author1= |edition=illustrated |publisher=Nelson Thornes |year=1996 |isbn=978-0-17-431439-4 |page=26 |url=https://books.google.com/books?id=HpaBTnefqnkC}} [https://books.google.com/books?id=HpaBTnefqnkC&pg=PA26 Extract of page 26]</ref>

Because the quadratic equation involves only one unknown, it is called "[[univariate]]". The quadratic equation contains only [[exponentiation|powers]] of {{math|''x''}} that are non-negative integers, and therefore it is a [[polynomial equation]]. In particular, it is a [[degree of a polynomial|second-degree]] polynomial equation, since the greatest power is two.