Quadratic Equations & Cubic Equation Formula

The simple version of the quadratic formula was used 2000 years back by Babylonian mathematicians. The equation was almost the same as we are using today and it was written by a Hindu mathematician named Brahmagupta. This was only the quadratic equation that defined the concept of imaginary numbers and how can you show the same over a graph. This is useful for a variety of applications in science, engineering, advanced mathematics, construction, machine manufacturing and many more.

Quadratic Equations Formula

You must be surprised to know quadratic equations are a crucial part of our daily lives. Take an example of swing that is mobbing back and forth. When it is moving continuously, what type of shape will you notice? Obviously, this is a sort of arch or a part of the circle. Yes, of course!

The particular shape that is formed by swing movement can be understood better with quadratic equation when you graph it. In mathematics, quadratic equation is an expression with highest degree two and it is written in the form as given below –

\[\large Quadratic\;Equation\;=ax^{2} + bx + c = 0 \]

Where a, b, c are three numbers that could be written as zero. This would be interesting to know that each quadratic equation has two solutions either real or imaginary. Next is the quadratic formula that is needed to compute the roots for a quadratic equation. Further, it would be interesting to know the history of quadratic formula how can you use this formula and prove it?

Every quadratic equation has two solutions that are calculated with the help of quadratic formula. In just a few simple steps, this is possible to find the solution either it is a whole number, rational number, or an imaginary number. In mathematics, the quadratic formula is given as –

\[x = {-b \pm \sqrt{b^2-4ac} \over 2a}.\]

Equation Formula

For the polynomial having a degree two is called the quadratic equation that means it is squared. In mathematics, the standard quadratic equation formula is given as –

For each quadratic equation with degree two, there are two solutions that may be real or imaginary. This may be easy to solve quadratic equations with the help of quadratic formulas but to make them useful in daily application, you must have a depth understanding of the program. They are also needed to prepare yourself for the competitive exams.

Cubic Equation Formula

The cubic equation has either one real root or it may have three-real roots. For the polynomial having a degree three is known as the cubic polynomial. In mathematics, the cubic equation formula can be given as –

\[\LARGE ax^{3}+bx^{2}+cx+d=0\]

Depressing the Cubic Equation

Substitute \[\large x= y-\frac{b}{3a}\] in the above cubic equation, then we get,

\[\large a\left ( y-\frac{b}{3a} \right )^{3}+b\left ( y-\frac{b}{3a} \right )^{2}+c\left ( y-\frac{b}{3a} \right )+d=0\]

Simplifying further, we obtain the following depressed cubic equation –

\[\large ay^{3}+\left ( c-\frac{b^{2}}{3a} \right )y+\left ( d+\frac{2b^{3}}{27a^{2}} +\frac{bc}{3a}\right )=0\]

It must have the term in x3 or it would not be cubic ( and so a≠0), but any or all of b, c and d can be zero. For instance:

For instance: \[\large x^{3}-6\times 2+11x-6=0\;or\;4x^{3}+57=0\;or\;x^{3}+9x=0\]

Exponential Equation Formula

The exponential equation is the equation where each side can be represented with the same base and it can be solved with the help of property. It can also be used to design a graph for compound interest, radioactive decay, and growth of population etc. In mathematics, the exponential equation formula can be given as –

\[\large Exponential\;Equation = y=ab^{x}\]

Where x and y are the constants and a, be would be the constants.

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