Earliest methods used to solve quadratic equations were geometric. Babylonian cuneiform tablets from around 1800-1600 BCE contain problems that can be reduced to solving quadratic equations, showing they understood techniques. The Egyptians also solved quadratic equations geometrically in the Middle Kingdom around 2050-1650 BCE. Later mathematicians like Euclid, Brahmagupta, and al-Khwārizmī developed more algebraic methods, with Brahmagupta explicitly describing the quadratic formula around 628 AD. The need for convenience ultimately led to the discovery of the general quadratic formula, first obtained by Simon Stevin in 1594 and published by René Descartes in 1637 in the modern form still used today.

1. How people solve quadratic equations during the ancient time
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 (2050 BC to 1650 BC), contains the solution to a two-term
quadratic equation.
The Greek mathematician Euclid (circa 300 BC) 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 (597–668 AD) 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.[17]
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.
Earliest Methods used to solve Quadratic Equations
Methods used by the Babylonians
Babylonian mathematics (also known as Assyro-Babylonian mathematics[1][2][3][4][5][6]
) was any
mathematics developed or practiced by the people of Mesopotamia, from the days of the
early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful
and well edited.[7]
In respect of time they fall in two distinct groups: one from the Old
Babylonian period (1830-1531 BC), the other mainly Seleucid from the last three or four
centuries BC. In respect of content there is scarcely any difference between the two groups of
texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two
millennia.[7]

2. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge
of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s.
Written in Cuneiform script, tablets were inscribed while the clay was moist, and baked hard in
an oven or by the heat of the sun. The majority of recovered clay tablets date from 1800 to 1600
BCE, and cover topics that include fractions, algebra, quadratic andcubic equations and
the Pythagorean theorem. The Babylonian tablet YBC 7289 gives an approximation to √2
accurate to three sexagesimal places (seven significant digits).
Origins of Babylonian mathematics
Babylonian mathematics is a range of numeric and more advanced mathematical practices in
the ancient Near East, written in cuneiform script. Study has historically focused on the Old
Babylonian period in the early second millennium BC due to the wealth of data available. There
has been debate over the earliest appearance of Babylonian mathematics, with historians
suggesting a range of dates between the 5th and 3rd millennia BC. Babylonian mathematics was
primarily written on clay tablets in cuneiform script in the Akkadian or Sumerianlanguages.
Babylonian numerals
The Babylonian system of mathematics was sexagesimal (base 60) numeral system. From this
we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360
degrees in a circle.[citation needed]
The Babylonians were able to make great advances in
mathematics for two reasons. Firstly, the number 60 is a superior highly composite number,
having factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 (including those that are themselves
composite), facilitating calculations with fractions. Additionally, unlike the Egyptians and
Romans, the Babylonians had a true place-value system, where digits written in the left column
represented larger values (much as in our base ten system: 734 = 7×100 + 3×10 + 4×1). The
Sumerians and Babylonians were pioneers in this respect.

3. Thomas Carlyles Geometric Solution
In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with
a quadratic equation. The circle has the property that the solutions of the quadratic equation are
the horizontal coordinates of the intersections of the circle with the horizontal axis.[1]
The idea of
using such a circle to solve a quadratic equation is attributed to Thomas Carlyle (1795–
1881).[2]
Carlyle circles have been used to develop ruler-and-compass constructions of regular
polygons.
Carlyle circle of the quadratic equation x2
− sx + p = 0.
Given the quadratic equation
x2
− sx + p = 0
the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as
a diameter is called the Carlyle circle of the quadratic equation.
Pythagorean Geometric Solution
In mathematics, the Pythagorean theorem—or Pythagoras's theorem—is a relation
in Euclidean geometry among the three sides of a right triangle. It states that the square of
the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other
two sides. Thetheorem can be written as an equation relating the lengths of the sides a, b and c,
often called the Pythagorean equation:
where c represents the length of the hypotenuse, and a and b represent the lengths of the other
two sides.

4. Discription Of Its Methods
The Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—
ca. 495 BC), who by tradition is credited with itsproof,[2][3]
although it is often argued that
knowledge of the theorem predates him. There is evidence that Babylonian
mathematicians understood the formula, although there is little surviving evidence that they used
it in a mathematical framework.[4][5]
Mesopotamian, Indian and Chinese mathematicians have all
been known for independently discovering the result, some even providing proofs of special
cases.
The theorem has numerous proofs, possibly the most of any mathematical theorem. These are
very diverse, including both geometric proofs and algebraic proofs, with some dating back
thousands of years. The theorem can be generalized in various ways,
including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not
right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The
Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical
abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals,
songs, stamps and cartoons abound.

5. Project in
Mathematics
Submitted by:
Charish Kaye Radaña
Submitted to:
Tr. Jeanette Umbay