![]() The general form of a cubic is, after dividing by the leading coefficient,Īs with the quadratic equation, there are several forms for the cubic when negative terms are moved to the other side of the equation and zero terms dropped.īack in the 16th century it was a big deal to solve cubic equations. It seems to take a lot of time before people will extend their concept of number to include new entities.Įquations of the third degree are called cubic equations. One would think that the consolidation of four cases into one might be enough justification for accepting negative numbers, but apparently it wasn't. ![]() With hindsight, we see that the 15th century solutions are just special cases of the quadratic formula. Each of these forms required a different form of a solution. ![]() There are other forms, but either they have no solutions among the positive numbers or else they can be reduced to linear equations. This one form gives rise to four forms when you move the negative terms to the other side of the equation and when you drop zero terms from the equation: Since the leading coefficient a is not zero in a quadratic equation, you can always divide by it to get an equivalent quadratic equation where a equals 1, that is, x 2 + bx + c = 0. Instead, quadratic equations were classified into four different kinds depending on the signs of the coefficients a, b, and c. Where there are two distinct real solutions if the discriminant b 2 4 ac is positive, one double real solution if the discriminant is 0, and no real solutions if the discriminant is negative.īack in the 15th century, this was not understood. We also know that the general solution is given by the quadratic formula: With negative numbers we understand that every quadratic equation in the variable x can be written in the form (Some ancient cultures, including that of China and India, accepted negative numbers, but not the ones mentioned above.) Negative numbers were not yet accepted as entities. ![]() In all this mathematics, only positive numbers were considered to be numbers. At that time works in mathematics were translated from the Arabic into Latin allowing Western European scholars to learn about the medieval Arabic-language mathematics and the older Greek mathematics, such as Euclid's Elements. Mathematics reawakened in Western Europe in the 13th century. ![]()
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