This article is about algebraic equations of degree two and their solutions. For the formula used to find solutions to such equations, see Quadratic formula. For functions defined by polynomials of degree two, see Quadratic function.
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as[1]
where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.[2]
The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression 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 numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates 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 factored into an equivalent equation[3]
where r and s are the solutions for x.
The quadratic formula
expresses the solutions in terms of a, b, and 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.[4][5]
Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.
^Charles P. McKeague (2014). Intermediate Algebra with Trigonometry (reprinted ed.). Academic Press. p. 219. ISBN 978-1-4832-1875-5. Extract of page 219
^Protters & Morrey: "Calculus and Analytic Geometry. First Course".
^The Princeton Review (2020). Princeton Review SAT Prep, 2021: 5 Practice Tests + Review & Techniques + Online Tools. Random House Children's Books. p. 360. ISBN 978-0-525-56974-9. Extract of page 360
^David Mumford; Caroline Series; David Wright (2002). Indra's Pearls: The Vision of Felix Klein (illustrated, reprinted ed.). Cambridge University Press. p. 37. ISBN 978-0-521-35253-6. Extract of page 37
^Mathematics in Action Teachers' Resource Book 4b (illustrated ed.). Nelson Thornes. 1996. p. 26. ISBN 978-0-17-431439-4. Extract of page 26
and 26 Related for: Quadratic equation information
In mathematics, a quadraticequation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as a x 2 + b x + c = 0 , {\displaystyle...
zero, then the result is a quadraticequation. The solutions of a quadraticequation are the zeros of the corresponding quadratic function. The bivariate...
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the second degree, or equations or formulas that involve such terms. Quadratus is Latin for square. Quadratic function (or quadratic polynomial), a polynomial...
numbers first arise in the teaching of quadraticequations and the quadratic formula. For example, the quadraticequation x 2 + x + 1 = 0 {\displaystyle x^{2}+x+1=0}...
BC could solve some kinds of quadraticequations (displayed on Old Babylonian clay tablets). Univariate algebraic equations over the rationals (i.e., with...
a_{0}z^{2}+a_{2}z+a_{4}=0\,\!} which is a simple quadraticequation, whose solutions are easily found using the quadratic formula: z = − a 2 ± a 2 2 − 4 a 0 a 4...
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tangent line at that point, or as the solution of certain bivariate quadraticequations such as the reciprocal relationship x y = 1. {\displaystyle xy=1...
not to be confused with a quadraticequation, which has only one variable and includes terms of degree two or less. A quadratic form is one case of the...
The treatise provided for the systematic solution of linear and quadraticequations. According to one history, "[i]t is not certain just what the terms...
This is typically the case when considering polynomial equations, such as quadraticequations. However, for some problems, all variables may assume either...
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linear equation for degree one quadraticequation for degree two cubic equation for degree three quartic equation for degree four quintic equation for degree...
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