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(Redirected from Quadratic formula)
In mathematics, a 'quadratic equation' is a polynomial equation of the second degree. The general form is
:
where ''a'' ≠ 0. (For ''a'' = 0, the equation becomes a linear equation.)
The letters ''a'', ''b'', and ''c'' are called coefficients: the quadratic coefficient ''a'' is the coefficient of ''x''2, the linear coefficient ''b'' is the coefficient of ''x'', and ''c'' is the constant coefficient, also called the free term or constant term.
Quadratic equations are called ''quadratic'' because ''quadratus'' is Latin for "square"; in the leading term the variable is squared.
A quadratic equation with real (or complex) coefficients has two (not necessarily distinct) solutions, called ''roots'', which may be real or complex, given by the 'quadratic formula':
:
where the symbol "±" indicates that both
:{|
|-
|
| style="width:100px" align="center" | and
|
|}
are solutions.
In the above formula, the expression underneath the square root sign:
:
is called the ''discriminant'' of the quadratic equation.
A quadratic equation with ''real'' coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:
★ If the discriminant is positive, there are two distinct roots, both of which are real numbers. For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers—in other cases they may be quadratic irrationals.
★ If the discriminant is zero, there is exactly one root, and that root is a real number. Sometimes called a double root, its value is:
★ :
★ If the discriminant is negative, there are ''no'' real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other:
★ :
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
The roots of the quadratic equation
:
are also the zeros of the quadratic function:
:
since they are the values of ''x'' for which
:
If ''a'', ''b'', and ''c'' are real numbers, and the domain of ''f'' is the set of real numbers, then the zeros of ''f'' are exactly the ''x''-coordinates of the points where the graph touches the ''x''-axis.
It follows from the above that, if the discriminant is positive, the graph touches the ''x''-axis at two points, if zero, the graph touches at one point, and if negative, the graph does not touch the ''x''-axis.
The term
:
is a factor of the polynomial
:
if and only if ''r'' is a root of the quadratic equation
:
It follows from the quadratic formula that
:
In the special case where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as
:
Certain higher-degree equations may be 'quadratic in form', such as:
:
which can be written as
:
where
: .
Note that the highest exponent is twice the value of the exponent of the middle term. This equation may be resolved directly or with a simple substitution, using the methods that are available for the quadratic, such as factoring, the quadratic formula, or completing the square.
Generally speaking, if the polynomial is quadratic in some variable ''u'' where
:
then the quadratic equation can be used to help find solutions.
The Babylonians, as early as 1800 BC (displayed on Old Babylonian clay tablets) could solve a pair of simultaneous equations of the form:
:
which are equivalent to the equation:[1]
:
The original pair of equations were solved as follows:
#Form
#Form
#Form
#Form
#Find by inspection of the values in (1) and (4). Stillwell, John. 2004. ''Mathematics and its History''. Berlin and New York: Springer-Verlag. 542 pages. p. 87
In the Sulba Sutras in ancient India circa 8th century BCE quadratic equations of the form ''ax''2 = ''c'' and ''ax''2 + ''bx'' = ''c'' were explored using geometric methods. Babylonian mathematicians from circa 400 BCE and Chinese mathematicians from circa 200 BCE used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BCE.
In 628 CE, Brahmagupta gave the first explicit (although still not completely general) solution of the quadratic equation:
:
This is equivalent to:
:
The ''Bakhshali Manuscript'' dated to have been written in India in the 7th century CE contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ''ax''/''c'' = ''y''). Mohammad bin Musa Al-kwarismi (Persia, 9th century) developed a set of formulae that worked for positive solutions. Abraham bar Hiyya Ha-Nasi (also known by the Latin name Savasorda) introduced the complete solution to Europe in his book ''Liber embadorum'' in the 12th century. Bhāskara II (1114–1185), an Indian mathematician–astronomer, gave the first general solution to the quadratic equation with two roots.[2]
The writing of the Chinese mathematician Yang Hui (1238-1298 AD) represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.
The quadratic formula can be derived by the method of completing the square, so as to make use of the algebraic identity:
:
Dividing the quadratic equation
:
by ''a'' (which is allowed because ''a'' is non-zero), gives:
:
or
:
The quadratic equation is now in a form in which the method of completing the square can be applied.
To "complete the square" is to find some constant ''k'' such that
:
for another constant ''y''. In order for these equations to be true,
:
and
:
thus
::
Adding this constant to equation (1) produces
:
The left side is now a perfect square because
:
The right side can be written as a single fraction, with common denominator 4''a''2. This gives
:
Taking the square root of both sides yields
:
Isolating ''x'', gives
:
Start with the general form of a quadratic:
:
Multiply both sides by 4''a'':
:
Subtract 4''ac'' from both sides:
:
Add ''b''2 to both sides:
:
Factorise the left-hand side:
:
Take the square root of both sides:
:
Subtract ''b'' from both sides:
:
Divide both sides by 2''a'':
:
In some situations it is preferable to express the roots in an alternate form.
:
This alternative requires ''c'' to be nonzero; for, if ''c'' is zero, the formula correctly gives zero as one root, but fails to give any second, non-zero root. (When ''c'' is zero we have division of zero by zero, which is indeterminate.)
The roots are the same regardless of which expression we use; the alternate form is merely an algebraic variation of the common form:
:
A careful floating point computer implementation differs a little from both forms to produce a robust result. Assuming the discriminant, ''b''2−4''ac'', is positive and ''b'' is nonzero, the code will be something like the following.
:
:
:
Here sgn(''b'') is the sign function, where sgn(''b'') is 1 if ''b'' is positive and −1 if ''b'' is negative; its use ensures that the quantities added are of the same sign, avoiding ''catastrophic cancellation''. The computation of ''r''2 uses the fact that the product of the roots is ''c''/''a''.
Viète's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form:
:
and
:
The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, when there are two real roots the vertex’s x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is given by the expression:
:
The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving
:
The formula and its derivation remain correct if the coefficients ''a'', ''b'' and ''c'' are complex numbers, or more generally members of any field whose characteristic is not 2. (In a field of characteristic 2, the element 2''a'' is zero and it is impossible to divide by it.)
The symbol
:
in the formula should be understood as "either of the two elements whose square is
:
if such elements exist. In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Note that even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.
In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial
:
over a field of characteristic 2. If ''b'' = 0, then the solution reduces to extracting a square root, so the solution is
:
and note that there is only one root since
:
In summary,
:
See quadratic residue for more information about extracting square roots in finite fields.
In the case that ''b'' ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the '2-root' ''R''(''c'') of ''c'' to be a root of the polynomial ''x''2 + ''x'' + ''c'', an element of the splitting field of that polynomial. One verifies that ''R''(''c'') + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ''ax''2 + ''bx'' + ''c'' are
:
and
:
For example, let ''a'' denote a multiplicative generator of the group of units of 'F'4, the Galois field of order four (thus ''a'' and ''a'' + 1 are roots of ''x''2 + ''x'' + 1 over 'F'4). Because (''a'' + 1)2 = ''a'', ''a'' + 1 is the unique solution of the quadratic equation ''x''2 + ''a'' = 0. On the other hand, the polynomial ''x'' + ''ax'' + 1 is irreducible over 'F'4, but splits over 'F'16, where it has the two roots ''ab'' and ''ab'' + ''a'', where ''b'' is a root of ''x''2 + ''x'' + ''a'' in 'F'16.
This is a special case of Artin-Schreier theory.
1. Stillwell, John. 2004. ''Mathematics and its History''. Berlin and New York: Springer-Verlag. 542 pages. p. 86
2. http://www.bbc.co.uk/dna/h2g2/A2982567
''Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas'', by Swami Sankaracarya (1884-1960), Motilal Banarsidass Indological Publishers and Booksellers, Varnasi, India, 1965; reprinted in Delhi, India, 1975, 1978. 367 pages.
★ Linear equation
★ Cubic equation
★ Quartic equation
★ Quintic equation
★ Fundamental theorem of algebra
★ Parabola
★ Quadratic function
★ Solving quadratic equations with continued fractions
★ Periodic points of complex quadratic mappings
★ Chakravala method
★
★ L. Euler's Elements of Algebra
★ Quadratic equation solver, plus solvers for cubic and quartic equations
★ 101 uses of a quadratic equation part I Part II
★ Quadratic graphical explorer Interactive applet. Sliders for a,b,c show effects on a graph.
In mathematics, a 'quadratic equation' is a polynomial equation of the second degree. The general form is
:
where ''a'' ≠ 0. (For ''a'' = 0, the equation becomes a linear equation.)
The letters ''a'', ''b'', and ''c'' are called coefficients: the quadratic coefficient ''a'' is the coefficient of ''x''2, the linear coefficient ''b'' is the coefficient of ''x'', and ''c'' is the constant coefficient, also called the free term or constant term.
Quadratic equations are called ''quadratic'' because ''quadratus'' is Latin for "square"; in the leading term the variable is squared.
Plots of real-valued quadratic function ''ax''2 + ''bx'' + ''c'', varying each coefficient separately
Quadratic formula
A quadratic equation with real (or complex) coefficients has two (not necessarily distinct) solutions, called ''roots'', which may be real or complex, given by the 'quadratic formula':
:
where the symbol "±" indicates that both
:{|
|-
|
| style="width:100px" align="center" | and
|
|}
are solutions.
Discriminant
Example discriminant signs
/>■ <0: ''x''2+1⁄2
/>■ <0: ''x''2+1⁄2
In the above formula, the expression underneath the square root sign:
:
is called the ''discriminant'' of the quadratic equation.
A quadratic equation with ''real'' coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:
★ If the discriminant is positive, there are two distinct roots, both of which are real numbers. For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers—in other cases they may be quadratic irrationals.
★ If the discriminant is zero, there is exactly one root, and that root is a real number. Sometimes called a double root, its value is:
★ :
★ If the discriminant is negative, there are ''no'' real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other:
★ :
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
Geometry
For the quadratic function:
''f'' (''x'') = ''x''2 − ''x'' − 2 = (''x'' + 1)(''x'' − 2) of a real variable ''x'', the ''x''-coordinates of the points where the graph touches the ''x''-axis, ''x'' = −1 and ''x'' = 2, are the roots of the quadratic equation: ''x''2 − ''x'' − 2 = 0.
''f'' (''x'') = ''x''2 − ''x'' − 2 = (''x'' + 1)(''x'' − 2) of a real variable ''x'', the ''x''-coordinates of the points where the graph touches the ''x''-axis, ''x'' = −1 and ''x'' = 2, are the roots of the quadratic equation: ''x''2 − ''x'' − 2 = 0.
The roots of the quadratic equation
:
are also the zeros of the quadratic function:
:
since they are the values of ''x'' for which
:
If ''a'', ''b'', and ''c'' are real numbers, and the domain of ''f'' is the set of real numbers, then the zeros of ''f'' are exactly the ''x''-coordinates of the points where the graph touches the ''x''-axis.
It follows from the above that, if the discriminant is positive, the graph touches the ''x''-axis at two points, if zero, the graph touches at one point, and if negative, the graph does not touch the ''x''-axis.
Quadratic factorization
The term
:
is a factor of the polynomial
:
if and only if ''r'' is a root of the quadratic equation
:
It follows from the quadratic formula that
:
In the special case where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as
:
Application to higher-degree equations
Certain higher-degree equations may be 'quadratic in form', such as:
:
which can be written as
:
where
: .
Note that the highest exponent is twice the value of the exponent of the middle term. This equation may be resolved directly or with a simple substitution, using the methods that are available for the quadratic, such as factoring, the quadratic formula, or completing the square.
Generally speaking, if the polynomial is quadratic in some variable ''u'' where
:
then the quadratic equation can be used to help find solutions.
History
The Babylonians, as early as 1800 BC (displayed on Old Babylonian clay tablets) could solve a pair of simultaneous equations of the form:
:
which are equivalent to the equation:[1]
:
The original pair of equations were solved as follows:
#Form
#Form
#Form
#Form
#Find by inspection of the values in (1) and (4). Stillwell, John. 2004. ''Mathematics and its History''. Berlin and New York: Springer-Verlag. 542 pages. p. 87
In the Sulba Sutras in ancient India circa 8th century BCE quadratic equations of the form ''ax''2 = ''c'' and ''ax''2 + ''bx'' = ''c'' were explored using geometric methods. Babylonian mathematicians from circa 400 BCE and Chinese mathematicians from circa 200 BCE used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BCE.
In 628 CE, Brahmagupta gave the first explicit (although still not completely general) solution of the quadratic equation:
:
This is equivalent to:
:
The ''Bakhshali Manuscript'' dated to have been written in India in the 7th century CE contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ''ax''/''c'' = ''y''). Mohammad bin Musa Al-kwarismi (Persia, 9th century) developed a set of formulae that worked for positive solutions. Abraham bar Hiyya Ha-Nasi (also known by the Latin name Savasorda) introduced the complete solution to Europe in his book ''Liber embadorum'' in the 12th century. Bhāskara II (1114–1185), an Indian mathematician–astronomer, gave the first general solution to the quadratic equation with two roots.[2]
The writing of the Chinese mathematician Yang Hui (1238-1298 AD) represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.
Derivation
Complete the square
The quadratic formula can be derived by the method of completing the square, so as to make use of the algebraic identity:
:
Dividing the quadratic equation
:
by ''a'' (which is allowed because ''a'' is non-zero), gives:
:
or
:
The quadratic equation is now in a form in which the method of completing the square can be applied.
To "complete the square" is to find some constant ''k'' such that
:
for another constant ''y''. In order for these equations to be true,
:
and
:
thus
::
Adding this constant to equation (1) produces
:
The left side is now a perfect square because
:
The right side can be written as a single fraction, with common denominator 4''a''2. This gives
:
Taking the square root of both sides yields
:
Isolating ''x'', gives
:
Alternative derivation
Start with the general form of a quadratic:
:
Multiply both sides by 4''a'':
:
Subtract 4''ac'' from both sides:
:
Add ''b''2 to both sides:
:
Factorise the left-hand side:
:
Take the square root of both sides:
:
Subtract ''b'' from both sides:
:
Divide both sides by 2''a'':
:
Alternative formula
In some situations it is preferable to express the roots in an alternate form.
:
This alternative requires ''c'' to be nonzero; for, if ''c'' is zero, the formula correctly gives zero as one root, but fails to give any second, non-zero root. (When ''c'' is zero we have division of zero by zero, which is indeterminate.)
The roots are the same regardless of which expression we use; the alternate form is merely an algebraic variation of the common form:
:
Floating point implementation
A careful floating point computer implementation differs a little from both forms to produce a robust result. Assuming the discriminant, ''b''2−4''ac'', is positive and ''b'' is nonzero, the code will be something like the following.
:
:
:
Here sgn(''b'') is the sign function, where sgn(''b'') is 1 if ''b'' is positive and −1 if ''b'' is negative; its use ensures that the quantities added are of the same sign, avoiding ''catastrophic cancellation''. The computation of ''r''2 uses the fact that the product of the roots is ''c''/''a''.
Viète's formulas
Viète's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form:
:
and
:
The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, when there are two real roots the vertex’s x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is given by the expression:
:
The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving
:
Generalizations
The formula and its derivation remain correct if the coefficients ''a'', ''b'' and ''c'' are complex numbers, or more generally members of any field whose characteristic is not 2. (In a field of characteristic 2, the element 2''a'' is zero and it is impossible to divide by it.)
The symbol
:
in the formula should be understood as "either of the two elements whose square is
:
if such elements exist. In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Note that even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.
Characteristic 2
In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial
:
over a field of characteristic 2. If ''b'' = 0, then the solution reduces to extracting a square root, so the solution is
:
and note that there is only one root since
:
In summary,
:
See quadratic residue for more information about extracting square roots in finite fields.
In the case that ''b'' ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the '2-root' ''R''(''c'') of ''c'' to be a root of the polynomial ''x''2 + ''x'' + ''c'', an element of the splitting field of that polynomial. One verifies that ''R''(''c'') + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ''ax''2 + ''bx'' + ''c'' are
:
and
:
For example, let ''a'' denote a multiplicative generator of the group of units of 'F'4, the Galois field of order four (thus ''a'' and ''a'' + 1 are roots of ''x''2 + ''x'' + 1 over 'F'4). Because (''a'' + 1)2 = ''a'', ''a'' + 1 is the unique solution of the quadratic equation ''x''2 + ''a'' = 0. On the other hand, the polynomial ''x'' + ''ax'' + 1 is irreducible over 'F'4, but splits over 'F'16, where it has the two roots ''ab'' and ''ab'' + ''a'', where ''b'' is a root of ''x''2 + ''x'' + ''a'' in 'F'16.
This is a special case of Artin-Schreier theory.
References
1. Stillwell, John. 2004. ''Mathematics and its History''. Berlin and New York: Springer-Verlag. 542 pages. p. 86
2. http://www.bbc.co.uk/dna/h2g2/A2982567
Book
''Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas'', by Swami Sankaracarya (1884-1960), Motilal Banarsidass Indological Publishers and Booksellers, Varnasi, India, 1965; reprinted in Delhi, India, 1975, 1978. 367 pages.
See also
★ Linear equation
★ Cubic equation
★ Quartic equation
★ Quintic equation
★ Fundamental theorem of algebra
★ Parabola
★ Quadratic function
★ Solving quadratic equations with continued fractions
★ Periodic points of complex quadratic mappings
★ Chakravala method
External links
★
★ L. Euler's Elements of Algebra
★ Quadratic equation solver, plus solvers for cubic and quartic equations
★ 101 uses of a quadratic equation part I Part II
★ Quadratic graphical explorer Interactive applet. Sliders for a,b,c show effects on a graph.
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