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

ax² + 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- 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- 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 ax2 + bx + c = 0, as follows:

x+ (b / a) x = - c / a

If we look at the first term (to the left of the = sign), we see that the binomial “ax2+ (b / a) x ”is missing a term to be a perfect square trinomial (a+ 2ab + b2). This term is the square of half the coefficient of the second term, that is, (b / 2a)2 or what is the same, b2/ (4th2):

x+ (b / a) x + b2/ (4th2) = - 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 , b2/ (4th2). 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+ (b / a) x + b2/ (4th2) = - c / a + b2/ (4th2)

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

(x + b / 2a)= - c / a + b2/ (4th2)

Extracting the square root of both members we have:

√ (x + b / 2a)2 = ± √ [- c / a + b2/ (4th2) ]

(x + b / 2a) = ± √ [- c / a + b2/ (4th2) ]

Consequently, the roots of the quadratic formula will be:

•  x= -b / 2a + √ [- c / a + b2/ (4th2) ]
• x= -b / 2a - √ [- c / a + b2/ (4th2) ]