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An equation is a statement in mathematics, in which the left hand side is equal to the right hand side. All equations in mathematics have variables in them. An equation in which the variable has the highest power of 2 is called a quadratic equation in mathematics.
Thus, we can define quadratic equation as an equation that has a degree of 2.
A quadratic equation is differentiated from other equations by means of the degree. The degree of an equation is the highest exponent on any variable present in that equation. Sometimes you have to simplify a given mathematical equation before concluding wether it is quadratic or not.

An example of a quadratic equation is x^2 = 25.
Here, there is a right hand side, a left hand side and an equal to sign in between. Furthermore, the variable x in this equation has the highest power of 2, thus the degree of the equation is 2.

A quadratic equation typically has two solutions, that is, we obtain two values for x after solving a quadratic equation. For example in the equation x^2 = 25, taking square root of both sides, we get x = sqrt(25). Now we know that the square root of 25 is 5, but the extended concept of quadratics says that even if we multiply -5 with -5, we will get 25, and thus -5 is also a square root of 25. Thus there are two solutions for the above quadratic equation: one is 5, and the other is -5.

It is to be noted that the two solutions of a quadratic equation can also be equal. For example, the solutions of the quadratic equation x^2 - 4x + 4 = 0 are both  x = 2. Then also it is said that the quadratic equation has two solutions, although equal.

The solutions of a quadratic equation have different names. They are also called roots, zeroes and x-intercepts.