When we find a quadratic or quadratic equation, we see that it has the following form:

**ax****² + bx + c = 0**

where «x» is the variable or unknown, and the letters a, b and c are the coefficients, which can have any value, except that a = 0. To find a solution to this type of equation, we use the **quadratic formula** which has the following form:

**x = [- b ± √ (b ^{2 }- 4ac)] / 2a**

Substituting the values of the coefficients a, b and c into it, we can easily obtain the values of x, remembering that «±» expresses that the equation has!**TWO SOLUTIONS**! Part "b^{2 }- 4ac ”is called discriminant and:

- if it is positive, there are TWO solutions.
- if it is zero there is only ONE solution.
- if it is negative there are two solutions that include imaginary numbers.

**Demonstration**

Let's rewrite the quadratic equation ax^{2} + bx + c = 0, as follows:

x^{2 }+ (b / a) x = - c / a

If we look at the first term (to the left of the = sign), we see that the binomial “ax^{2}+ (b / a) x ”is missing a term to be a perfect square trinomial (a^{2 }+ 2ab + b^{2}). This term is the square of half the coefficient of the second term, that is, (b / 2a)^{2} or what is the same, b^{2}/ (4th^{2}):

x^{2 }+ (b / a) x + b^{2}/ (4th^{2}) = - c / a

Indeed, a trinomial has been formed whose first term is the square of x, its second is twice the product of x times b / 2a, and its third is the square of half the coefficient of the second term b / 2a, that is , b^{2}/ (4th^{2}).

Now, so as not to alter the equation, we add to the second member the same amount that we have added to the first, so we will have:

x^{2 }+ (b / a) x + b^{2}/ (4th^{2}) = - c / a + b^{2}/ (4th^{2})

In the first member of the equation we have a perfect square trinomial, therefore:

(x + b / 2a)^{2 }= - c / a + b^{2}/ (4th^{2})

Extracting the square root of both members we have:

√ (x + b / 2a)^{2} = ± √ [- c / a + b^{2}/ (4th^{2}) ]

**(x + b / 2a) = ± √ [- c / a + b ^{2}/ (4th^{2}) ]**

Consequently, the roots of the quadratic formula will be:

- x
_{1 }= -b / 2a + √ [- c / a + b^{2}/ (4th^{2}) ] - x
_{2 }= -b / 2a - √ [- c / a + b^{2}/ (4th^{2}) ]