“Formula for Quadratic equation” these words come to your mind when you start to solve Quadratic equations as a new topic in your study. As we know quad means double that’s why one variable in the Quadratic equation is based on squared. Basically, the word quad is a Latin word and when we solve the quadratic equation, we find it in its standard form as \(ax^2 + bx + c=0\) The most important method to solve quadratic equations is the Quadratic formula \(x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{ 2a }\) The solution of the quadratic equation by this formula is very easy. We just put the values of a, b, & c in the formula then solve the equation according to the structure of the quadratic formula and get required solution sets. In this modern & scientific era online
Quadratic formula calculator is also available to solve the quadratic equations in minutes with the accuracy of the result. Now we discuss who invented the Quadratic formula? or the history of the Quadratic equation formula.
History Of The Formula For Quadratic Equation
Babylonians Contributions:
Through the Babylonian’s clay tablets as early as 2000 BC the Babylonian mathematicians used quadratic equations by geometric method to find out the area related problems of rectangles. They solved the quadratic equation only for the positive roots and the process which they used to find the values of x and y was as under:
First, they calculated the half value of p.
Secondly, they got the square of the result.
In the third step they subtracted the value of q.
In the fourth step they found the positive square root of the values.
Finally, they added the result of the first and 4th step to get the value of x.
The above-mentioned steps may be written as in the form of formula:
\(x = \dfrac{p}{2} + \sqrt{(\dfrac{p}{2})^2-q}\)
The Babylonian mathematicians didn’t use the geometrical method as in the form of formula but if we compare the above-mentioned formula with the today formula as \(x=\dfrac{-b+\sqrt{b^2-4ac}}{2a} \) then in the today formula \(a=1, b=-p, \& \space c = q \space \) So, we can say that it became the origin of the formula for the quadratic equation.
solve the quadratic equation by the quadratic formula \(x^2 – 6x + 8 = 0\)
Solution:
Here \(a=1,b=-6, \space \& \space c=8\)
Now apply the quadratic formula:
\(x = \dfrac{-b\pm\sqrt{b^2-4ac}}{ 2}\)
\(x=\dfrac{-(-6)\pm\sqrt{(-6)^2-4(4)(8)}}{2(1)} \)
\(x=\dfrac{6\pm\sqrt{36-32 }}{ 2}\)
\(x=\dfrac{6\pm\sqrt{4 }}{ 2}\)
\(x = \dfrac{ 6 \pm 2}{2} \)
\(x = \dfrac{ 6 + 2 } { 2 } \) OR \( x = \dfrac{ 6 – 2}{2}\)
\(x = \dfrac{8} { 2} \) OR \( x = \dfrac{4} {2} \)
\(x = 4 \) OR \( x = 2\)
Solution sets {4, 2}. Answer
EXAMPLE#2:
Solve the quadratic equation \(4x^2 + 12x = 7 \) by the Quadratic formula.
Solution:
First, we equation as: \(4x^2 + 12 x – 7 = 0 \)
Now solve the equation here \(a = 4, b = 12, \space \& \space c = – 7 \)
Apply the quadratic formula:
\(x = \dfrac{- b \pm \sqrt{b^2 – 4ac}}{ 2a}\)
\(x = \dfrac{ – 12 \pm \sqrt{(12)^2 – 4 (4) (-7) }}{ 2 (4)}\)
Users Of The Geometric Method To Solve The Quadratic Equations:
Besides Babylonian mathematicians, the mathematicians of China, Egypt, India and Greece also used the geometric method (not in the form of general formula) to solve the quadratic equations only for the positive roots.
Contribution Of Indian Mathematician:
In 628 AD an Indian mathematician named Brahmagupta explained in wording the following explicit form of formula to solve the quadratic equations as:
\(x = \dfrac{\sqrt{4ac + b^2 } \space – b }{ 2a}\)
Contributions Of Al Khawarizimi And Other Muslim Mathematicians:
In 9th century an Islamic mathematician Muhammad ibn Musa al-Khwarizmi inspired by the concept of Brahmagupta and developed a set of formula to solve the positive roots of the quadratic equations. He also developed the completing square method and identified that the discriminant should be positive in solving quadratic equations. Another Muslim mathematician Hamid ibn Turk proved the working of the Khwarizmi that the discriminant should not be negative to solve the quadratic equations.
The common thing in the above discussion is that all the mathematicians solved the quadratic equations for positive roots but later on Muslim mathematicians proved that the quadratic equations could be solved for negative roots as well as for irrational numbers.
Contributions Of Simon Stevin And Rene Descartes:
Finally in 1594 Simon Stevin by his work obtained the formula for Quadratic equation and in 1637 the formula for Quadratic equation which we use today as \(x = \dfrac{ – b \pm \sqrt{b^2 – 4ac }}{ 2a } \) was published by Rene Descartes in La Geometrie.
So, the above complete history of the quadratic formula is telling us how the quadratic formula derived. The quadratic formula is the best Quadratic equation solver. There are other methods to solve the quadratic equations like completing the square method, factoring method and graphing method etc. but the quadratic formula is widely used by the people of different fields to solve the quadratic equations problems. Now we understand the Quadratic formula’s calculation by the following examples:
EXAMPLE #1:
\(x = \dfrac{ – 12 \pm \sqrt{144 + 112 }}{ 8}\)
\(x = \dfrac{ – 12 \pm \sqrt{256 }}{ 8}\)
\(x = \dfrac{ – 12 \pm 16 }{ 8}\)
\(x = \dfrac{ – 12 + 16 }{ 8} \) OR \( x = \dfrac{ – 12 = 16 }{ 8} \)
\(x = \dfrac{4} {8} \) OR \( x = \dfrac{ – 28 } {8} \)
\(x = \dfrac{1 }{ 2} \) OR \( x = \dfrac{ – 7 } {2}\)
Solution sets {0.5, – 3.5} Answer.
From the above discussion and Quadratic formula calculation examples it will be clear to you how the Quadratic formula derived? And how you can solve the quadratic equation by Quadratic formula.
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