Deriving the Quadratic Formula via Completing the Square

The quadratic equation is a fundamental concept in algebra, particularly when solving equations of the form (ax^2 bx c 0). One method to derive the solution for this equation is by completing the square. This method not only provides a clear understanding of the solution but also lays the groundwork for the quadratic formula. In this article, we will explore the step-by-step process of deriving the quadratic formula using the method of completing the square.

What is Completing the Square?

Completing the square is a technique used to rewrite a quadratic equation in a form that allows us to easily find its roots. This method involves transforming the equation into a perfect square trinomial, which can be factored as the square of a binomial.

Deriving the Quadratic Formula

Let's start with the generic quadratic equation:

1. Isolate the constant term:
(ax^2 bx c 0)
Subtract (c) from both sides:
(ax^2 bx -c)

2. Set the coefficient of the (x^2) term to 1:
Divide both sides by (a):
(x^2 frac{b}{a}x -frac{c}{a})

3. Complete the square:
To complete the square, we need to add a specific quantity to both sides of the equation to form a perfect square trinomial on the left-hand side. This quantity is (left(frac{b}{2a}right)^2):

(x^2 frac{b}{a}x left(frac{b}{2a}right)^2 -frac{c}{a} left(frac{b}{2a}right)^2)

4. Simplify the left-hand side:
(x^2 frac{b}{a}x left(frac{b}{2a}right)^2) can be written as (left(x frac{b}{2a}right)^2):

(left(x frac{b}{2a}right)^2 -frac{c}{a} left(frac{b}{2a}right)^2)

5. Simplify the right-hand side:
On the right-hand side, combine the terms over a common denominator:

(left(x frac{b}{2a}right)^2 frac{b^2}{4a^2} - frac{c}{a}) Multiply the second term by (frac{4a}{4a}) to get a common denominator:

(left(x frac{b}{2a}right)^2 frac{b^2 - 4ac}{4a^2})

6. Take the square root of both sides:
Remember to consider both the positive and negative roots:

(x frac{b}{2a} pm frac{sqrt{b^2 - 4ac}}{2a})

7. Isolate (x):
Subtract (frac{b}{2a}) from both sides:

(x -frac{b}{2a} pm frac{sqrt{b^2 - 4ac}}{2a})

Pull the terms together into a single fraction:

(x boxed{frac{-b pm sqrt{b^2 - 4ac}}{2a}})

Example

Let's solve the quadratic equation (x^2 - 5x 6 0):

1. Isolate the constant term:
(x^2 - 5x -6)

2. Set the coefficient of the (x^2) term to 1:
This equation is already in the correct form.

3. Complete the square:
Add (left(frac{-5}{2}right)^2 frac{25}{4}) to both sides:

(x^2 - 5x frac{25}{4} -6 frac{25}{4})

4. Simplify the right-hand side:
(-6 frac{25}{4} frac{-24}{4} frac{25}{4} frac{1}{4})

(left(x - frac{5}{2}right)^2 frac{1}{4})

5. Take the square root of both sides:
(left(x - frac{5}{2}right) pm frac{1}{2})

6. Isolate (x):
(x frac{5}{2} pm frac{1}{2})