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]

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.

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.

If the discriminant is positive, then there are two distinct roots

Solving quadratic equations with continued fractions

Linear equation

Cubic function

Quartic equation

Quintic equation

Fundamental theorem of algebra

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Quadratic equation"

"Quadratic equations". MathWorld.

Weisstein, Eric W.

Archived 2007-11-10 at the Wayback Machine

101 uses of a quadratic equation

Archived 2007-10-22 at the Wayback Machine