Quadratic equation

In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as^[1] $ax^{2}+bx+c=0\,,$ where $x$ represents an unknown value, and $a$ , $b$ , and $c$ represent known numbers, where $a \neq 0$ . (If $a = 0$ and $b \neq 0$ 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] $ax^{2}+bx+c=a(x-r)(x-s)=0$ where $r$ and $s$ are the solutions for $x$ .

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

[1] Charles P. McKeague (2014). Intermediate Algebra with Trigonometry (reprinted ed.). Academic Press. p. 219. ISBN 978-1-4832-1875-5. Extract of page 219

[2] Protters & Morrey: "Calculus and Analytic Geometry. First Course".

[3] 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

[4] 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

[5] Mathematics in Action Teachers' Resource Book 4b (illustrated ed.). Nelson Thornes. 1996. p. 26. ISBN 978-0-17-431439-4. Extract of page 26

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Quadratic equation

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