A quadratic equation or quadratic equation can have zero, one or two real solutions, depending on the coefficients that appear in said equation. If you work on complex numbers, then you can say that every quadratic equation has two solutions.

To begin with, a quadratic equation is an equation of the form ax²+bx+c=0, where a, b, and c are real numbers and x is a variable.

x1 is said to be a solution of the above quadratic equation if replacing x with x1 satisfies the equation, that is, if a(x1)²+b(x1)+c=0.

If we have, for example, the equation x²-4x+4=0, then x1=2 is the solution, since (2)²-4(2)+4=4-8+4=0.

Conversely, if x2=0 is substituted, we obtain (0)²-4(0)+4=4 and since 4≠0 then x2=0 is not a solution of the quadratic equation.

**Solutions of a quadratic equation**

The number of solutions of a quadratic equation can be separated into two cases that are:

**1.-** **in real numbers**

When working with real numbers, quadratic equations can have:

**-Zero solutions:** that is, there is no real number that satisfies the quadratic equation. For example, the equation given the equation x²+1=0, there is no such real number that satisfies said equation, since both x² is greater than or equal to zero and 1 is strictly greater than zero, so that their sum will be greater strict than zero.

**-A repeated solution:** there is only one real value that satisfies the quadratic equation. For example, the only solution to the equation x²-4x+4=0 is x1=2.

**-Two different solutions:** There are two values that satisfy the quadratic equation. For example, x²+x-2=0 has two distinct solutions that are x1=1 and x2=-2.

**2.- In complex numbers**

When working with complex numbers, quadratic equations always have two solutions, which are z1 and z2 where z2 is the conjugate of z1. In addition, they can be classified into:

**-Complexes:** the solutions are of the form z=p±qi, where p and q are real numbers. This case corresponds to the first case in the list above.

**-Pure Complexes:** it is when the real part of the solution is equal to zero, that is to say that the solution has the form z=±qi, where q is a real number. This case corresponds to the first case in the list above.

**-Complexes with imaginary part equal to zero:** it is when the complex part of the solution is equal to zero, that is to say that the solution is a real number. This case corresponds to the last two cases in the previous list.

**How are the solutions of a quadratic equation calculated?**

To calculate the solutions of a quadratic equation, a formula known as «the resolvent» is used, which says that the solutions of an equation ax²+bx+c=0 are given by the expression in the following image:

The quantity that appears inside the square root is called the discriminant of the quadratic equation and is denoted by the letter “d”.

The quadratic equation will have:

-Two real solutions if, and only if, d>0.

-A repeated real solution if, and only if, d=0.

-Zero real solutions (or two complex solutions) if and only if d<0.

**examples**

-The solutions of the equation x²+x-2=0 are given by:

-The equation x²-4x+4=0 has a repeated solution which is given by:

-The solutions of the equation x²+1=0 are given by:

As can be seen in this last example, x2 is the conjugate of x1.