# Quadratic equation

**quadratic equationsquadraticquadratic formulasecond degreeABC formulaFactoring a quadratic expressionquadratic functionsquadratic modelquadratic problemquadratic term**

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form : wherewikipedia

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### Quadratic formula

Completing the square is the standard method for that, which results in the quadratic formula, which express the solutions in terms of a, b, and c. Graphing may also be used for getting an approximate value of the solutions.

In elementary algebra, the quadratic formula is a formula which provides the solution(s) to a quadratic equation.

### Algebra

**algebraicAlgebra IAlgebra 1**

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form

For example, in the quadratic equation

### Completing the square

**complete the square**

Completing the square is the standard method for that, which results in the quadratic formula, which express the solutions in terms of a, b, and c. Graphing may also be used for getting an approximate value of the solutions.

Completing the square may be used to solve any quadratic equation.

### Real number

**realrealsreal-valued**

If there is no real solution, there are two complex solutions.

850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.

### Algebraic equation

**polynomial equationalgebraic equationspolynomial equations**

that are non-negative integers, and therefore it is a polynomial equation.

The study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000 BC could solve some kinds of quadratic equations (displayed on Old Babylonian clay tablets).

### Imaginary unit

**isquare root of minus onesquare root of −1**

is the imaginary unit.

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation

### Quadratic irrational number

**quadratic irrationalQuadratic surdirrational quadratic numbers**

In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the set of rational numbers.

### Factorization

**factoringfactorfactors**

:Thus, the process of solving a quadratic equation is also called factorizing or factoring.

However, even for solving quadratic equations, factoring method was not used before Harriot’s work published in 1631, ten years after his death.

### Complex conjugate

**complex conjugationconjugateconjugation**

If the parabola does not intersect the x-axis, there are two complex conjugate roots.

According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as the quadratic equation or the cubic equation), so is its conjugate.

### Square root

**square rootssquareradical**

:Taking the square root of both sides, and isolating

It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry.

### Plus-minus sign

**±plus or minusPlus or minus sign**

The plus-minus symbol "±" indicates that both

:describing the two solutions to the quadratic equation ax 2 + bx + c = 0.

### Equation solving

**solutionsolutionsroot**

The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of its left-hand side.

:x 6 − 5x 3 + 6 = 0,(by using the substitution x = z 1/3, which simplifies this to a quadratic equation in z).

### Conic section

**conicconic sectionsconics**

The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.

In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables.

### Descartes' theorem

**Descartes theoremDescartes' Circle EquationDescartes' theorem (4 tangent circles)**

Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.

In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation.

### Square (algebra)

**squaresquaredsquares**

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form

### Babylonian mathematics

**BabyloniansBabylonian mathematiciansBabylonian**

Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve problems relating the areas and sides of rectangles.

The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem.

### Muller's method

A lesser known quadratic formula, as used in Muller's method provides the same roots via the equation

We do not use the standard formula for solving quadratic equations because that may lead to loss of significance.

### Loss of significance

**catastrophic cancellationcancellationdifferences of similar values**

In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root.

For example, consider the quadratic equation

### Nested radical

**nested radicalsLandau's algorithmcan be solved**

The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation.

:It follows by Vieta's formulas that x and y must be the roots of the quadratic equation

### Muhammad ibn Musa al-Khwarizmi

**Al-Khwarizmial-KhwārizmīMuhammad ibn Mūsā al-Khwārizmī**

). Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions.

Al-Khwarizmi's popularizing treatise on algebra (The Compendious Book on Calculation by Completion and Balancing, c. 813–833 CE ) presented the first systematic solution of linear and quadratic equations.

### Golden ratio

**golden sectiongolden meanφ**

The golden ratio is found as the positive solution of the quadratic equation x^2-x-1=0.

However, this is no special property of, because polynomials in any solution x to a quadratic equation can be reduced in an analogous manner, by applying:

### Abraham bar Hiyya

**Abraham bar Hiyya Ha-NasiAbraham bar ḤiyyaAbraham bar Chiia**

The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation.

A Hebrew treatise on practical geometry and Islamic algebra, the book contains the first known complete solution of the quadratic equation, and influenced the work of Leonardo Fibonacci.

### Hyperbola

**hyperbolicrectangular hyperbolahyperbolas**

The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.

where λ 1 and λ 2 are the roots of the quadratic equation

### Arithmetica

In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.

Most of the Arithmetica problems lead to quadratic equations.