BERDA BARLAK
QUADRATIC EQUATIONS AND POLYNOMIALS
When you see letters mixed with numbers and
arithmetic, like x +2x-7, there is a good chance that it is
2
a polynomial. Mathematicians, scientists, and engineers use
polynomials to solve problems. Polynomials are taught
in algebra, which is a gateway course to all technical
subjects.
The earliest known use of the equal sign is in Robert
Recorde's The Whetstone of Witte, 1557. The signs + for
addition, − for subtraction, and the use of a letter for an
unknown appear in Michael Stifel's Arithemetica integra,
1544. René Descartes, in La géometrie, 1637, introduced the
concept of the graph of a polynomial equation. He
popularized the use of letters from the beginning of the
alphabet to denote constants and letters from the end of
the alphabet to denote variables, as can be seen above, in
the general formula for a polynomial in one variable,
where the a 's denote constants and x denotes a variable.
Descartes introduced the use of superscripts to denote
exponents as well.
Polynomials are often used to form
polynomial equations, such as x +2x-7 = 0, or
2
polynomial functions, such as f(x) = x +2x-7.
2
……….
Throughout the years, the history of mathematics has
taken its fair share of changes. It has stretched across the
world from the Far East, migrating into the Western
Hemisphere. One of the most fundamental and key principles
of mathematics has been the quadratic formula. Having
been used in several different cultures, the formula has
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been part of the base of mathematics theory. The general
equation has been derived from many different sources,
most commonly: ax2 + bx + c = 0, with x being the variable
and a, b, and c its respective constant terms. Though this is
how modern mathematics perceives the equation, different
symbols and notations have been used to represent the
formula.
Beginning in the “Before Christ” era, the BaBylonians were
the first to have been recorded demonstrating the equation,
circa 400 BC. The form most mathematics students use today
is:
To solve a quadratic equation the Babylonians essentially
used the standard formula, with the a term being included in
the x2 variable. They considered two types of quadratic
equations, namely:
Here b and c were positive but not necessarily integers
Proof
The quadratic formula is proved by completing the square,
Divide the quadratic equation by a :
move
Use the method of completing the square
To "complete the square" is to find some "k" so that:
for some y.
and
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so
Add
equation:
to both sides of the
makes:
The left side is now a perfect
square; it is the square of
The right side can be a single
fraction, with a common
denominator 4a .
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Find the square root of
both sides.
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