We will explain how completing the square is used to derive the Quadratic Formula. We can consider a quadratic equation written in standard form, complete the square, and derive the formula by ourselves. Let's start by writing the constant term c on the right-hand side and dividing the equation by the leading coefficient a.
Next, we want the left-hand side of the above equation to be a perfect square trinomial. For this to happen, we need three terms. The first and third terms must be perfect squares and the second term must be twice the product of the square root of the first and third terms. To achieve this, we will add ( b2a)^2 to both sides.
Note that in one of the denominators we have the expression sqrt(a^2). Depending on the value of a, we can simplify this in two ways.
sqrt(a^2) = a, ifa≥0 - a, ifa<0
However, since we are considering both the positive and negative signs for the term containing this expression, we can simplify sqrt(a^2) as a.
Starting from a quadratic equation written in standard form, by completing the square, we have deduced the Quadratic Formula.
ax^2+bx+c=0 ⇔ x=- b± sqrt(b^2-4ac)/2a
