In basic algebra, the quadratic formula is
the solution of the quadratic equation.
There are other ways to solve the quadratic
equation instead of using the quadratic formula,
such as factoring, completing the square,
or graphing.
However, using the quadratic formula is often
the most convenient way.
The general quadratic equation is
Here x represents an unknown, and a, b, and
c are constants with a not equal to 0.
One can verify that the quadratic formula
satisfies the quadratic equation, by inserting
the former into the latter.
Each of the solutions given by the quadratic
formula is called a root of the quadratic
equation.
Derivation of the formula
Once a student understands how to complete
the square, they can then derive the quadratic
formula.
For that reason, the derivation is sometimes
left as an exercise for the student, who can
thereby experience rediscovery of this important
formula.
The explicit derivation is as follows.
Divide the quadratic equation by a, which
is allowed because a is non-zero:
Subtract c/a from both sides of the equation,
transforming it into the form
The quadratic equation is now in a form to
which the method of completing the square
can be applied.
To "complete the square", add a constant to
both sides of the equation such that the left
hand side becomes a complete square:
which produces
or
The square has thus been completed, as shown
in the figure.
Taking the square root of both sides yields
Isolating x gives the quadratic formula:
The plus-minus symbol "±" indicates that
both
are solutions of the quadratic equation.
There are many alternatives of this derivation
with minor differences, mostly concerning
the manipulation of .
Some sources, particularly older ones, use
alternative parameterizations of the quadratic
equation such as or , where b has a magnitude
one half of the more common one.
These result in slightly different forms for
the solution, but are otherwise equivalent.
Historical development
The earliest methods for solving quadratic
equations were geometric.
Babylonian cuneiform tablets contain problems
reducible to solving quadratic equations.
The Egyptian Berlin Papyrus, dating back to
the Middle Kingdom, contains the solution
to a two-term quadratic equation.
The Greek mathematician Euclid used geometric
methods to solve quadratic equations in Book
2 of his Elements, an influential mathematical
treatise.
Rules for quadratic equations appear in the
Chinese The Nine Chapters on the Mathematical
Art circa 200 BC.
In his work Arithmetica, the Greek mathematician
Diophantus solved quadratic equations with
a method more recognizably algebraic than
the geometric algebra of Euclid.
However, his solution gave only one root,
even when both roots were positive.
The Indian mathematician Brahmagupta explicitly
described the quadratic formula in his treatise
Brāhmasphuṭasiddhānta published in 628
AD, but written in words instead of symbols.
His solution of the quadratic equation was
as follows: "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."
This is equivalent to:
The 9th century Persian mathematician al-Khwārizmī,
influenced by earlier Greek and Indian mathematicians,
solved quadratic equations algebraically.
Mathematician Elizabeth Stapel has explained
that the need for convenience motivated the
discovery of the formula.
The quadratic formula covering all cases was
first obtained by Simon Stevin in 1594.
In 1637 René Descartes published La Géométrie
containing the quadratic formula in the form
we know today.
The first appearance of the general solution
in the modern mathematical literature appeared
in an 1896 paper by Henry Heaton.
Importance of this solution
Among the many equations that one encounters
while studying algebra, the quadratic formula
is one of the most important, and is considered
the most useful method of solving quadratic
equations.
Unlike some other solution methods such as
factoring, the quadratic formula can be used
to solve any quadratic equation.
Many equations that do not initially appear
to be quadratic can be put into quadratic
form, and solved using the quadratic formula.
For these reasons, it is often memorized.
Completing the square also allows for the
solution of all quadratics, as it is mathematically
equivalent, but the quadratic formula gives
a result without the need for so much algebraic
manipulation.
As such, it is generally considered more practical
to use the formula.
However, completing the square is very useful
for other purposes, such as putting the equations
for conic sections into standard form.
Other derivations
A number of alternative derivations of the
quadratic formula can be found in the literature.
These derivations either are simpler than
the standard completing the square method,
represent interesting applications of other
frequently used techniques in algebra, or
offer insight into other areas of mathematics.
Alternate method of completing the square
The great majority of algebra texts published
over the last several decades teach completing
the square using the sequence presented earlier:
divide each side by a, rearrange, then add
the square of one-half of b/a.
However, as pointed out by Larry Hoehn in
1975, completing the square can be accomplished
by a different sequence that leads to a simpler
sequence of intermediate terms: multiply each
side by 4a, rearrange, then add .
In other words, the quadratic formula can
be derived as follows:
This actually represents an ancient derivation
of the quadratic formula, and was known to
the Hindus at least as far back as 1025 AD.
Compared with the derivation in standard usage,
this alternate derivation is shorter, involves
fewer computations with literal coefficients,
avoids fractions until the last step, has
simpler expressions, and uses simpler math.
As Hoehn states, "it is easier 'to add the
square of b' than it is 'to add the square
of half the coefficient of the x term'".
By substitution
Another technique is solution by substitution.
In this technique, we substitute into the
quadratic to get:
Expanding the result and then collecting the
powers of produces:
We have not yet imposed a second condition
on and , so we now choose m so that the middle
term vanishes.
That is, or . Subtracting the constant term
from both sides of the equation and then dividing
by a gives:
Substituting for gives:
Therefore ; substituting provides the quadratic
formula.
By using algebraic identities
Let the roots of the standard quadratic equation
be and . At this point, we recall the identity:
Taking square root on both sides, we get
Since the coefficient a ≠ 0, we can divide
the standard equation by a to obtain a quadratic
polynomial having the same roots.
Namely,
From this we can see that the sum of the roots
of the standard quadratic equation is given
by , and the product of those roots is given
by
Hence the identity can be rewritten as:
Now,
Since, , if we take then we obtain and if
we instead take then we calculate that Combining
these results by using the standard shorthand,
we have that the solutions of the quadratic
equation are given by:
By Lagrange resolvents
An alternative way of deriving the quadratic
formula is via the method of Lagrange resolvents,
which is an early part of Galois theory.
This method can be generalized to give the
roots of cubic polynomials and quartic polynomials,
and leads to Galois theory, which allows one
to understand the solution of algebraic equations
of any degree in terms of the symmetry group
of their roots, the Galois group.
This approach focuses on the roots more than
on rearranging the original equation.
Given a monic quadratic polynomial
assume that it factors as
Expanding yields
where and .
Since the order of multiplication does not
matter, one can switch and and the values
of p and q will not change: one says that
p and q are symmetric polynomials in and . In
fact, they are the elementary symmetric polynomials
– any symmetric polynomial in and can be
expressed in terms of and The Galois theory
approach to analyzing and solving polynomials
is: given the coefficients of a polynomial,
which are symmetric functions in the roots,
can one "break the symmetry" and recover the
roots?
Thus solving a polynomial of degree n is related
to the ways of rearranging n terms, which
is called the symmetric group on n letters,
and denoted For the quadratic polynomial,
the only way to rearrange two terms is to
swap them, and thus solving a quadratic polynomial
is simple.
To find the roots and consider their sum and
difference:
These are called the Lagrange resolvents of
the polynomial; notice that one of these depends
on the order of the roots, which is the key
point.
One can recover the roots from the resolvents
by inverting the above equations:
Thus, solving for the resolvents gives the
original roots.
Formally, the resolvents are called the discrete
Fourier transform of order 2, and the transform
can be expressed by the matrix with inverse
matrix The transform matrix is also called
the DFT matrix or Vandermonde matrix.
Now is a symmetric function in and so it can
be expressed in terms of p and q, and in fact
as noted above.
But is not symmetric, since switching and
yields . Since is not symmetric, it cannot
be expressed in terms of the polynomials p
and q, as these are symmetric in the roots
and thus so is any polynomial expression involving
them.
However, changing the order of the roots only
changes by a factor of and thus the square
is symmetric in the roots, and thus expressible
in terms of p and q.
Using the equation
yields
and thus
If one takes the positive root, breaking symmetry,
one obtains:
and thus
Thus the roots are
which is the quadratic formula.
Substituting yields the usual form for when
a quadratic is not monic.
The resolvents can be recognized as being
the vertex, and is the discriminant.
A similar but more complicated method works
for cubic equations, where one has three resolvents
and a quadratic equation relating and which
one can solve by the quadratic equation, and
similarly for a quartic equation, whose resolving
polynomial is a cubic, which can in turn be
solved.
However, the same method for a quintic equation
yields a polynomial of degree 24, which does
not simplify the problem, and in fact solutions
to quintic equations in general cannot be
expressed using only roots.
See also
Discriminant
Fundamental theorem of algebra
References
External links
Quadratic formula calculator
Quadratic formula calculator Online
Alternative formula
