Quadratic formula

quadratic formulaWhen 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).

Completing the square

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) ]