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Using the Quadratic Equation Solver

What is the Quadratic Equation Solver?

The quadratic equation solver is the calculator that helps to solve quadratic equations online quickly and efficiently. Quadratic equation is the equation of the second order, which looks like ax2+bx2+c=0, where a is not equal to 0. The value of the variable x is called the root of the quadratic equation, if when it is substituted, the equation becomes a true equality. Quadratic equation with real coefficients can have from 0 to 2 roots depending on the value of the discriminant. The roots can be both real and complex.

By using a quadratic equation solver available on any of the websites, you can solve quadratic equations and easily find the roots of a quadratic equation.

Quadratic equation solver will show the detailed solution to your example, which will help you understand the algorithm for solving problems and to remember the material studied in a classroom.

History of Quadratic Equations

Algebra originated in connection with the solution of various tasks using equations. Typically, tasks require you to find one or more unknown quantities. These tasks are about the solution of one equation or a system of several equations, or finding the unknown quantity by means of algebraic operations.

Some of the algebraic methods for solving linear and quadratic equations have been known 4,000 years ago in ancient Babylon. The need to solve equations in ancient times was caused by the need to solve the problems associated with finding squares of lands and land work, as well as with the development of astronomy and mathematics itself. As it was mentioned above, Babylonians could solve quadratic equations around 2000 B.C. While studying their cuneiform texts there were found both partial and full quadratic equations.

The rule for solving these equations in the Babylonian texts coincides substantially with a modern rule. It is unknown how the Babylonians found this rule.

Some methods of solving equations both quadratic and high degree equations were derived by the Arabs. A famous Arab mathematician Al-Khwarizmi in his book «Al-Jabbar» described many different ways to solve equations. Their feature was that Al-Khwarizmi used the complex radicals to find the roots of equations. The need for solving such equations was useful in matters of division of inheritance.

Quadratic equations were solved in India as well. Tasks for quadratic equations can be found in the astronomical treatise “Ariabhattiam” compiled in the year of 499 by the Indian mathematician and astronomer Ariabhattoy. Another Indian scientist Brahmagupta set out a general rule for solving quadratic equations representing them in a conical formula: ax²+bx=c, where a>0.

In this equation, coefficients, except a, may be negative as well. Brahmagupta’s rule is essentially the same as the one we have today.

In ancient India, public competitions in solving difficult tasks were very common.

Different equations both square equations and high degree equations were solved by our ancestors. These equations were solved in different and distant from each other countries. There was a large need in the equations. The equations were used in construction, military affairs, and in everyday situations.

In Europe, formulas solving quadratic equations based on al-Khwarizmi were first introduced in the “Book of the abacus”, written in 1202 by the Italian mathematician Leonardo Fibonacci. This voluminous work, which reflects the influence of mathematics, is different, competent, and clear. The author has developed some new algebraic examples of solving the tasks and was the first who introduced the negative numbers in Europe. His book contributed to the spread of algebraic knowledge not only in Italy but also in Germany, France and other European countries. Many tasks from the book were transferred in almost all European books of the 16th and 17th centuries.

The general rule for solving quadratic equations in its canonical form: X2+BX=C for all possible combinations of signs of the coefficients B, C, was formulated in Europe only in 1544 by Stiefel.

Vieta, who recognized only positive roots, has the derivation of the formula of solutions of the quadratic equation in general form. Italian mathematicians Tartaglia, Cardano, Bombelli were among the first in the 16th century who took into account both positive and negative roots. Only in the 17th century, thanks to the works of Girard, Descartes, Newton, and other scientists, a way of solving quadratic equations took a modern look.

Types of Quadratic Equations

What is the quadratic equation? How does it look like and how can it be calculated with a quadratic equation solver? «Square» is the key word in the term of a quadratic equation. It means that the equation must have a square value of X. Besides X, the equation might have or might not have the usual value of X (in the first degree) and just a number. The equation can’t have the degree that is more than two.

Mathematically speaking, the quadratic equation is the equation of the following form: AX2+BX+C=0, where A, B and C are some numbers. B and C can be absolutely any numbers, while A can be any number, except of 0. For example: X2+3X-4=0, where A=1; B=3; C=-4

  • The quadratic equations that have a full set of numbers on the left (X in the square with a coefficient A, X in the first degree with the coefficient B, and a free number) are called complete quadratic equations.
  • If B=0, X in the first degree will disappear. It happens when you multiply by zero. If C=0, there will be an equation without a constant number. If both coefficients, B and C, are equal to 0, everything is even easier: 2X2=0 or -0,3X2=0. Such equations, where something is missing, are called incomplete quadratic equations.

Please note that the square X is present in all the equations.

The A can’t be equal to 0, because if we put 0 instead of the A, the square X will disappear and the equation will become linear. It will need to be solved in a different way.

So the major types of quadratic equations are complete and incomplete.

Solving Quadratic Equations

One of the ways to solve the quadratic equations is using the quadratic equation solver. In this case, the solution of the quadratic equation has two stages.

  • First, the discriminant of the equation needs to be calculated using the following formula: D=B2-4AC.
  • On the quadratic equation solver the roots of a quadratic equation are calculated with the following formula: X1,2=(-B+-VD)/2A.

The other way of calculation quadratic equations besides using a quadratic equation solver is to apply formulas and clear simple rules. Quadratic equations are calculated easily. At first, it is necessary to bring the given equation to the standard form: AX2+BX+C=0. If you already have the equation in this form, then all you need to do is to properly identify all the coefficients A, B, and C.

The formula for finding the roots of a quadratic equation is the following: X1,2=(-B+-VB2-4AC)/2A. The expression under the root sign is called the discriminant. As you can see, to find X, you use only A, B, and C – the coefficients of the quadratic equation. Simply substitute values A, B, and C in the formula and calculate the equation.

In order to solve incomplete quadratic equations, you can also use the general formula. All you need to do is to correctly understand what is equal to A, B, and C.

If you don’t have a quadratic equation solver at hand, you need to know the formula of the quadratic equations by heart in order to solve the task.

Discriminant and Its Formula

Probably, everybody heard about discriminant. Discriminant is found in the most general formula for solving any quadratic equations, which is X1,2=(-B+-VB2-4AC)/2A. The expression under the root sign is called the discriminant. Usually discriminant is denoted by the letter D. Discriminant formula is the following: D=B2-4AC.

When solving a quadratic equation by this formula, there may be only three cases.

  • The discriminant is positive. This means the root can be extracted from it. No matter whether it is difficult to extract the root or not, the important thing is that it can be extracted in principle. In this case the quadratic equation has two roots and two different solutions.
  • The discriminant is equal to zero. Then there will be only one solution, because nothing changes when you add or subtract zero in the numerator. Strictly speaking, it is not one root, but two the same roots. But, in a simplified way it is said that there is one solution.
  • The discriminant is negative. The root can’t be extracted from a negative number. This means that there are no solutions.

Quadratic equations are widely used in solving trigonometric, exponential, logarithmic, irrational, and transcendental equations and inequalities. Quadratic equations play a huge role in the development of mathematics. It is important to know how to solve them with a quadratic equation solver and with general formulas.

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