Taming the Quadratic Formula
Here we are going to unravel the secrets behind one of the most essential tools in algebra: the quadratic formula. The quadratic formula has been the bane of many a math students’ first year algebra experience. It is difficult to memorize by rote and most algebra teachers don’t explain how it connects to the standard quadratic equation that students’ have already been studying.
Prerequisite; Complete the Square
The magic to deriving the quadratic formula is utilizing a technique called Completing The Square. What this does is
that it takes any quadratic expression and enables us to write it as perfect square (i.e. as a square of a binomial and plus or minus a constant). The idea is to add and subtract a term that will transform the quadratic equation into a perfect square trinomial so that we can then factor the expression down to the square of a binomial.
Are you ready? Let’s dive in!
Step 1: Start with a Quadratic Equation in Standard From
Let’s begin by considering a general quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants. Our goal is to derive a formula that gives us the solutions for x. In other words, we want an equation such that x= … something. That something is the Quadratic Formula.
{\large \text{1. } ax^{2} + bx + c = 0 }
Step 2: Subtract the Constant Term
First, let’s isolate the constant term, c, by subtracting it from both sides of the equation:
{\large \begin{aligned} ax^{2}+bx+c &= 0 \\ \text{2. } ax^{2}+bx &= -c \end{aligned} }
Step 3: Divide through by a
Divide both sides of the equation by a, the coefficient of x^2, which is what we call the leading coefficient.
{\large \begin{aligned} ax^{2} + bx &= -c \\ \text{3. }\frac{ax^{2} + bx}{a} &= \frac{-c}{a} \end{aligned} }
Step 4: Simplify the Left Side
Simplify the left side of the equation by dividing each term by a:
{\large \begin{aligned} \frac{ax^{2} + bx}{a} &= \frac{-c}{a} \\ \text{4. } x^{2} + \frac{bx}{a} &= \frac{-c}{a} \end{aligned} }
Step 5: Add \color{red}{(\frac{b}{2a})^2} to Both Sides
This is the magic sauce for completing the square we need to add the term \color{red}(\frac{b}{2a})^{2} to both sides of the equation. This is make the left side a perfect square and thus allow us to factor.
{\large \begin{aligned} x^{2} + \frac{bx}{a} &= \frac{-c}{a} \\ x^2 + \frac{b}{a}x &= \frac{-c}{a} \\ \text{5. }x^2 + \frac{b}{a}x \color{red}{+ (\frac{b}{2a})^{2}} &= \frac{-c}{a} \color{red}{+ (\frac{b}{2a})^{2}} \end{aligned} }
Step 6: Factor the Trinomial
Now the left hand side of the equation is a perfect square trinomial. It will factor following the following pattern;
x^{2} \pm bx + (\frac{b}{2})^{2} = (x \pm \frac{b}{2})^{2}.
{\large \begin{aligned}x^{2} + \frac{b}{a}x + (\frac{b}{2a})^{2} &= \frac{-c}{a} + (\frac{b}{2a})^{2} \\ \bm{\text{6. }}(x + \frac{b}{2a})^{2} &= \frac{-c}{a} + (\frac{b}{2a})^{2} \end{aligned} }
Step 7: Simplify
Now we should simplify the right side of the equation.
{\large \begin{aligned} (x + \frac{b}{2a})^{2} &= \frac{-c}{a} + (\frac{b}{2a})^{2} \\(x + \frac{b}{2a})^{2} &= \frac{-c}{a} + \frac{b^{2}}{4a^{2}} \\ (x + \frac{b}{2a})^{2} &= \frac{-4ac}{4a^{2}} + \frac{b^{2}}{4a^{2}} \\ (x + \frac{b}{2a})^{2} &= \frac{-4ac + b^{2}}{4a^{2}} \\ \text{7. }(x + \frac{b}{2a})^{2} &= \frac{b^{2}-4ac}{4a^{2}} \end{aligned} }
Step 8: Take the Square Root
Taking the square root of both sides will eliminate the square on the left side. Make sure the add the \pm outside the square root on the right side. This giving us:
{\large \begin{aligned} (x + \frac{b}{2a})^2 &= \frac{b^2-4ac}{4a^2} \\ \sqrt{(x + \frac{b}{2a})^{2}} &= \pm\sqrt{\frac{b^2-4ac}{4a^2}} \\ x + \frac{b}{2a} &= \pm\frac{\sqrt{b^2-4ac}}{\sqrt{4a^2}} \\ \text{8. }x + \frac{b}{2a} &= \pm\frac{\sqrt{b^2-4ac}}{2a} \end{aligned} }
Step 9: Isolate x
To solve for 
{\large \begin{aligned} x + \frac{b}{2a} &= \pm\frac{\sqrt{b^2-4ac}}{2a} \\ x &= \pm\frac{\sqrt{b^2-4ac}}{2a} - \frac{b}{2a} \\ \text{9. }x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \end{aligned} }
The Quadratic Formula
Congratulations! We have successfully derived the quadratic formula:
{\huge \color{red}{ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} }}
The Quadratic Formula… Demystified
You can also watch a video that I made on the steps to derive the quadratic formula, here;
