Why is the quadratic formula useful
In the case of a quadratic equation, the variable occurs in the second power and, if necessary, additionally in the first power; it must not be in the denominator. Every quadratic equation can be converted into the general form by equivalent transformations:
Here are , and arbitrary constants.
A quadratic equation has at most two solutions. How many and which solutions a quadratic equation has in a specific case can be determined directly if the equation is in the general form. The number of solutions is determined by the value of their so-called "discriminants" determined, which can be calculated directly using the general equation form (1). The following three cases can be distinguished:
This procedure, based on the discriminant Being able to infer the number and the values of the solutions is colloquially known as the “midnight formula”.   It can be applied to any quadratic equation that has the general form (1).
Special cases of quadratic equations
If there are special cases of quadratic equations, other, sometimes simpler solution methods can also be used:
Is , we have a quadratic equation of the following form:
This equation can be found directly after to be resolved:
The equation only has the above two solutions if and have different signs, otherwise the solution set is the same (if is) or equal to the empty set (if is).
The above equation can be clearly explained by the fact that for the square of every number always applies. If a square number is now multiplied by a positive factor, you cannot add another positive number to get the value zero as the result.
Is , so one is missing -free term, we have a quadratic equation of the following form:
In this case, the midnight formula provides the two values and as solutions. The same solutions are obtained by looking at the left hand side of the equation excluded as a common factor:
Since a product is only zero if (at least) one of the two factors is zero, it follows from the above equation that either the or must apply. It follows from the first case , follows from the second case (a linear equation) .
Product form of quadratic equations
are and the solutions of a quadratic equation, where also is permissible, this can generally also be represented in the following form:
Such a division of an equation into multiple linear factors is called linear factorization or product form. This representation plays only a minor role for quadratic equations, but it can also be used in a useful way for equations of a higher order.
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