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In mathematics, a quadratic equation is a polynomial equation of the second
degree. The general form is
ax^2+bx+c=0,\,\!
where a ≠ 0. (For if 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 x2, 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.
Quadratic formula
A quadratic equation with real or complex coefficients has two (not necessarily
distinct) solutions, called roots, which may or may not be real, given by the
quadratic formula:
x = \frac{-b \pm \sqrt {b^2-4ac}}{2a},
where the symbol "±" indicates that both
x_+ = \frac{-b + \sqrt {b^2-4ac}}{2a}\quad\textrm{and}\quad\ x_- = \frac{-b - \sqrt
{b^2-4ac}}{2a}
are solutions.
Discriminant
Example discriminant signs■ <0: x2+1?2■ =0: ?4?3x2+4?3x?1?3■ >0: 3?2x2+1?2x?4?3
Example discriminant signs
■ <0: x2+1?2
■ =0: ?4?3x2+4?3x?1?3
■ >0: 3?2x2+1?2x?4?3
In the above formula, the expression underneath the square root sign:
\Delta = b^2 - 4ac , \,\!
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 distinct root, and that root
is a real number. Sometimes called a double root, its value is:
x = -\frac{b}{2a} . \,\!
* 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:
\begin{align} x &= \frac{-b}{2a} + i \frac{\sqrt {4ac - b^2}}{2a} , \\ x &= \frac{-b}{2a}
- i \frac{\sqrt {4ac - b^2}}{2a} , \\ i^2 &= -1. \end{align}
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) = x2 ? x ? 2 = (x + 1)(x ? 2) of a real
variable x, the x-coordinates of the points where the graph intersects the
x-axis, x = ?1 and x = 2, are the roots of the quadratic equation: x2 ? x ? 2 =
0.
For the quadratic function:
f (x) = x2 ? x ? 2 = (x + 1)(x ? 2) of a real variable x, the x-coordinates of
the points where the graph intersects the x-axis, x = ?1 and x = 2, are the
roots of the quadratic equation: x2 ? x ? 2 = 0.
The roots of the quadratic equation
ax^2+bx+c=0,\,
are also the zeros of the quadratic function:
f(x) = ax^2+bx+c,\,
since they are the values of x for which
f(x) = 0.\,
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
x - r\,
is a factor of the polynomial
ax^2+bx+c, \
if and only if r is a root of the quadratic equation
ax^2+bx+c=0. \
It follows from the quadratic formula that
ax^2+bx+c = a \left( x - \frac{-b + \sqrt {b^2-4ac}}{2a} \right) \left( x - \frac{-b
- \sqrt {b^2-4ac}}{2a} \right).
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
ax^2+bx+c = a \left( x + \frac{b}{2a} \right)^2.\,\!
Application to higher-degree equations
Certain higher-degree equations can be brought into quadratic form and solved
that way. For example, the 6th-degree equation in x:
x^6 - 4x^3 + 8 = 0\,
can be rewritten as:
(x^3)^2 - 4(x^3) + 8 = 0\,,
or, equivalently, as a quadratic equation in a new variable u:
u^2 - 4u + 8 = 0,\,
where
u = x^3.\,
Solving the quadratic equation for u results in the two solutions:
u = 2 \pm 2i.
Thus
x^3 = 2 \pm 2i\,.
Concentrating on finding the three cube roots of 2 + 2i – the other three
solutions for x will be their complex conjugates – rewriting the right-hand side
using Euler's formula:
x^3 = 2^{\tfrac{3}{2}}e^{\tfrac{1}{4}\pi i} =
2^{\tfrac{3}{2}}e^{\tfrac{8k+1}{4}\pi i}\,
(since e2kπi = 1), gives the three solutions:
x = 2^{\tfrac{1}{2}}e^{\tfrac{8k+1}{12}\pi i}\,,~k = 0, 1, 2\,.
Using Eulers' formula again together with trigonometric identities such as cos(π/12)
= (√2 + √6) / 4, and adding the complex conjugates, gives the complete
collection of solutions as:
x_{1,2} = -1 \pm i,\,
x_{3,4} = \frac{1 + \sqrt{3}}{2} \pm \frac{1 - \sqrt{3}}{2}i\,
and
x_{5,6} = \frac{1 - \sqrt{3}}{2} \pm \frac{1 + \sqrt{3}}{2}i.\,
History
The Babylonians, as early as 1800 BC (displayed on Old Babylonian clay tablets)
could solve a pair of simultaneous equations of the form:
x+y=p,\ \ xy=q \
which are equivalent to the equation:
\ x^2+q=px
The original pair of equations were solved as follows:
1. Form \frac{x+y}{2}
2. Form \left(\frac{x+y}{2}\right)^2
3. Form \left(\frac{x+y}{2}\right)^2 - xy
4. Form \sqrt{\left(\frac{x+y}{2}\right)^2 - xy} = \frac{x-y}{2}
5. Find x,\ y by inspection of the values in (1) and (4).
In the Sulba Sutras in ancient India circa 8th century BCE quadratic equations
of the form ax2 = c and ax2 + 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:
\ ax^2+bx=c
“ To the absolute number multiplied by four times the [coefficient of the]
square, add the square of the [coefficient of the] middle term; the square root
of the same, less the [coefficient of the] middle term, being divided by twice
the [coefficient of the] square is the value. (Brahmasphutasiddhanta (Colebrook
translation, 1817, page 346) ”
This is equivalent to:
x = \frac{\sqrt{4ac+b^2}-b}{2a}.
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. His work was based on Brahmagupta. 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.
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
The quadratic formula can be derived by the method of completing the square, so
as to make use of the algebraic identity:
x^2+2xy+y^2 = (x+y)^2.\,\!
Dividing the quadratic equation
ax^2+bx+c=0 \,\!
by a (which is allowed because a is non-zero), gives:
x^2 + \frac{b}{a} x + \frac{c}{a}=0,\,\!
or
x^2 + \frac{b}{a} x= -\frac{c}{a} \qquad (1)
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
x^2 + \frac{b}{a}x + k = x^2+2xy+y^2,\,\!
for another constant y. In order for these equations to be true,
\frac{b}{a} = 2y\!
or
y = \frac{b}{2a}\,\!
and
k = y^2,\,\!
thus
k = \frac{b^2}{4a^2}.\,\!
Adding this constant to equation (1) produces
x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}.\,\!
The left side is now a perfect square because
x^2+\frac{b}{a}x+\frac{b^2}{4a^2} = \left( x + \frac{b}{2a} \right)^2
The right side can be written as a single fraction, with common denominator 4a2.
This gives
\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.
Taking the square root of both sides yields
\left|x+\frac{b}{2a}\right| = \frac{\sqrt{b^2-4ac\ }}{|2a|}\Rightarrow x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac\
}}{2a}.
Isolating x, gives
x=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac\ }}{2a}=\frac{-b\pm\sqrt{b^2-4ac\ }}{2a}.
Alternative formula
In some situations it is preferable to express the roots in an alternate form.
x =\frac{2c}{-b \mp \sqrt {b^2-4ac\ }} .
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.
Instead, one of the two choices for ? produces a division by zero, which is
undefined.
The roots are the same regardless of which expression we use; the alternate form
is merely an algebraic variation of the common form:
\begin{align} \frac{-b + \sqrt {b^2-4ac\ }}{2a} &{}= \frac{-b + \sqrt {b^2-4ac\
}}{2a} \cdot \frac{-b - \sqrt {b^2-4ac\ }}{-b - \sqrt {b^2-4ac\ }} \\ &{}=
\frac{4ac}{2a \left ( -b - \sqrt {b^2-4ac} \right ) } \\ &{}=\frac{2c}{-b - \sqrt
{b^2-4ac\ }}. \end{align}
The alternative formula can reduce loss of precision in the numerical evaluation
of the roots, which may be a problem if one of the roots is much smaller than
the other in absolute magnitude. The problem of c possibly being zero can be
avoided by using a mixed approach:
x_1 = \frac{-b - \sgn b \,\sqrt {b^2-4ac}}{2a},
x_2 = \frac{c}{ax_1}.
Here sgn denotes the sign function.
Floating point implementation
A careful floating point computer implementation differs a little from both
forms to produce a robust result. Assuming the discriminant, b2?4ac, is positive
and b is nonzero, the code will be something like the following.
t := -\tfrac12 \big( b + \sgn(b) \sqrt{b^2-4ac} \big) \,\!
r_{1} := t/a \,\!
r_{2} := c/t \,\!
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 r2 uses the fact that the
product of the roots is c/a.
See Numerical Recipes in C, Section 5.6: "Quadratic and Cubic Equations".
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:
x_+ + x_- = -\frac{b}{a}
and
x_+ \cdot x_- = \frac{c}{a}.
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:
x_V = \frac {x_+ + x_-} {2} = -\frac{b}{2a}.
The y-coordinate can be obtained by substituting the above result into the given
quadratic equation, giving
y_V = - \frac{b^2}{4a} + c = - \frac{ b^2 - 4ac} {4a}.
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 2a is zero and it is
impossible to divide by it.)
The symbol
\pm \sqrt {b^2-4ac}
in the formula should be understood as "either of the two elements whose square
is
b^2-4ac,\,
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
\displaystyle x^{2} + bx + c
over a field of characteristic 2. If b = 0, then the solution reduces to
extracting a square root, so the solution is
\displaystyle x = \sqrt{c}
and note that there is only one root since
\displaystyle -\sqrt{c} = -\sqrt{c} + 2\sqrt{c} = \sqrt{c}.
In summary,
\displaystyle x^{2} + c = (x + \sqrt{c})^{2}.
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 x2 + 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 ax2 + bx + c are
\frac{b}{a}R\left(\frac{ac}{b^2}\right)
and
\frac{b}{a}\left(R\left(\frac{ac}{b^2}\right)+1\right).
For example, let a denote a multiplicative generator of the group of units of
F4, the Galois field of order four (thus a and a + 1 are roots of x2 + x + 1
over F4). Because (a + 1)2 = a, a + 1 is the unique solution of the quadratic
equation x2 + a = 0. On the other hand, the polynomial x + ax + 1 is irreducible
over F4, but splits over F16, where it has the two roots ab and ab + a, where b
is a root of x2 + x + a in F16.
This is a special case of Artin-Schreier theory.
A linear equation is an algebraic equation in which each term is either a
constant or the product of a constant and (the first power of) a single
variable. Linear equations can have one, two, three or more variables. A common
form of a linear equation in the two variables x and y is
y = mx + b,\,
where m and b designate constants (the variable y is multiplied by the constant
1, which as usual is not explicitly written). The set of solutions of such an
equation forms a straight line in the plane, which is the origin of the name
"linear". In this particular equation, the constant m determines the slope or
gradient of that line; and the constant term b determines the point at which the
line crosses the y-axis.
Since terms of a linear equations cannot contain products of distinct or equal
variables, nor any power (other than 1) or other function of a variable,
equations involving terms such as xy, x2, y1/3, and sin(x) are nonlinear.
Forms for 2D linear equations
Complicated linear equations, such as the ones above, can be rewritten using the
laws of elementary algebra into several simpler forms. In what follows x, y and
t are variables; other letters represent constants (unspecified but fixed
numbers).
General form
Ax + By + C = 0,\,
where A and B are not both equal to zero. The equation is usually written so
that A ≥ 0, by convention. The graph of the equation is a straight line, and
every straight line can be represented by an equation in the above form. If A is
nonzero, then the x-intercept, that is the x-coordinate of the point where the
graph crosses the x-axis (y is zero), is ?C/A. If B is nonzero, then the
y-intercept, that is the y-coordinate of the point where the graph crosses the
y-axis (x is zero), is ?C/B, and the slope of the line is ?A/B.
Standard form
Ax + By = C,\,
where A, B, and C are integers whose greatest common factor is 1, A and B are
not both equal to zero and, A is non-negative (and if A = 0 then B has to be
positive). The standard form can be converted to the general form, but not
always to all the other forms if A or B is zero.
Slope–intercept form
Y-axis formula
y = mx + b,\,
where m is the slope of the line and b is the y-intercept, which is the
y-coordinate of the point where the line crosses the y axis. This can be seen by
letting x = 0, which immediately gives y = b.
X-axis formula
x = \frac{y}{m} + c,\,
where m ≠ 0, is the slope of the line and c is the x-intercept, which is the
x-coordinate of the point where the line crosses the x axis. This can be seen by
letting y = 0, which immediately gives x = c.
Point–slope form
y - y_1 = m \cdot ( x - x_1 ),
where m is the slope of the line and (x1,y1) is any point on the line. The
point-slope and slope-intercept forms are easily interchangeable.
The point-slope form expresses the fact that the difference in the y coordinate
between two points on a line (that is, y ? y1) is proportional to the difference
in the x coordinate (that is, x ? x1). The proportionality constant is m (the
slope of the line).
Intercept form
\frac{x}{c} + \frac{y}{b} = 1
where c and b must be nonzero. The graph of the equation has x-intercept c and
y-intercept b. The intercept form can be converted to the standard form by
setting A = 1/c, B = 1/b and C = 1.
Two-point form
y - k = \frac{q - k}{p - h} (x - h),
where p ≠ h. The graph passes through the points (h,k) and (p,q), and has slope
m = (q?k) / (p?h).
Parametric form
x = T t + U\,
and
y = V t + W.\,
Two simultaneous equations in terms of a variable parameter t, with slope m = V
/ T, x-intercept (VU?WT) / V and y-intercept (WT?VU) / T.
This can also be related to the two-point form, where T = p?h, U = h, V = q?k,
and W = k:
x = (p - h) t + h\,
and
y = (q - k)t + k.\,
In this case t varies from 0 at point (h,k) to 1 at point (p,q), with values of
t between 0 and 1 providing interpolation and other values of t providing
extrapolation.
Normal form
y \sin \phi + x \cos \phi - p = 0,\,
where φ is the angle of inclination of the normal and p is the length of the
normal. The normal is defined to be the shortest segment between the line in
question and the origin. Normal form can be derived from general form by
dividing all of the coefficients by
\frac{|C|}{-C}\sqrt{A^2 + B^2}.
This form also called Hesse standard form, named after a German mathematician
Ludwig Otto Hesse.
Special cases
y = b.\,
This is a special case of the standard form where A = 0 and B = 1, or of the
slope-intercept form where the slope M = 0. The graph is a horizontal line with
y-intercept equal to b. There is no x-intercept, unless b = 0, in which case the
graph of the line is the x-axis, and so every real number is an x-intercept.
x = c.\,
This is a special case of the standard form where A = 1 and B = 0. The graph is
a vertical line with x-intercept equal to c. The slope is undefined. There is no
y-intercept, unless c = 0, in which case the graph of the line is the y-axis,
and so every real number is a y-intercept.
y = y \ and x = x.\,
In this case all variables and constants have canceled out, leaving a trivially
true statement. The original equation, therefore, would be called an identity
and one would not normally consider its graph (it would be the entire xy-plane).
An example is 2x + 4y = 2(x + 2y). The two expressions on either side of the
equal sign are always equal, no matter what values are used for x and y.
e = f.\,
In situations where algebraic manipulation leads to a statement such as 1 = 0,
then the original equation is called inconsistent, meaning it is untrue for any
values of x and y (i.e. its graph would be the empty set) An example would be 3x
+ 2 = 3x ? 5.
Connection with linear functions and operators
In all of the named forms above (assuming the graph is not a vertical line), the
variable y is a function of x, and the graph of this function is the graph of
the equation.
In the particular case that the line crosses through the origin, if the linear
equation is written in the form y = f(x) then f has the properties:
f ( x + y ) = f ( x ) + f ( y )\,
and
f ( a x ) = a f ( x ),\,
where a is any scalar. A function which satisfies these properties is called a
linear function, or more generally a linear map. This property makes linear
equations particularly easy to solve and reason about.
Linear equations occur with great regularity in applied mathematics. While they
arise quite naturally when modeling many phenomena, they are particularly useful
since many non-linear equations may be reduced to linear equations by assuming
that quantities of interest vary to only a small extent from some "background"
state.
Linear equations in more than two variables
M System of linear equations
A linear equation can involve more than two variables. The general linear
equation in n variables is:
a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b.
In this form, a1, a2, …, an are the coefficients, x1, x2, …, xn are the
variables, and b is the constant. When dealing with three or fewer variables, it
is common to replace x1 with just x, x2 with y, and x3 with z, as appropriate.
Such an equation will represent an (n–1)-dimensional hyperplane in n-dimensional
Euclidean space (for example, a plane in 3-space).
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