Proof For The Quadratic Formula | Shreedharacharya's Method (With Explanations)

In our previous module, we saw what equations are and how to solve them. We also saw an interesting formula called the quadratic formula which can be used to solve quadratic equations easily.

Today, we going to see the proof or the derivation of the quadratic formula which is also known as the Shreedharacharya's method. Let's begin by observing a simple equation:

We know what a general form of a quadratic equation is, right? If you want to learn more about it, visit the previous module: What Are Equations? | Linear Equations | Quadratic Equations | Solving Equations.

Now, every quadratic equation is generalized as:

ax² + bx + c = 0, a ≠ 0; a, b and c are real numbers

We know that a is the coefficient of x² and b is the coefficient of x and c is the constant. Let's us now derive the quadratic formula.

ax² + bx + c = 0 [General Form]

Step 1: First, let us divide the whole equation by a. This gives:

x² + (bx / a) + (c / a) = 0

Now, why did we do this? If x² has coefficient 1, then it is easy for us to factorize. This step will come handy later on.

Step 2: Now, we take the constant (c / a) to the right hand side (RHS) of the equation. This gives:

x² + (bx / a) = - (c / a)

Again, why did we do this? This is to simplify the LHS of the equation so that by the end, we get only x² in the LHS and the rest of the quantity in the RHS of the equation.

Step 3: Next, we add (b / 2a)² to both the sides of the equation. This gives:

x² + (bx / a) + (b / 2a)² = - (c / a) + (b / 2a)²

This step might look absurd to you at first, but see what happens when we enter the next step. And by the way, this step does not affect our original equation that we started off at first. Adding, subtracting, multiplying or dividing the same quantities on both the sides of an equation does not affect or change it.

Step 4: We now rewrite the term (bx / a) as 2(bx / 2a). This gives:

x² + 2(bx / 2a) + (b / 2a)² = - (c / a) + (b / 2a)²

Can you see it? A PERFECT SQUARE! Yes, this is the reason why we added (b / 2a)² to both sides of the equation. 2(bx / 2a) is the same as (bx / a) since 2 gets cancelled off. We just multiplied 2 to the numerator as well as the denominator. So the quantity still remains the same. Now, this method of adding something to both sides of equation to form a perfect square is called "completing the square method".

Step 5: Completing the square. This gives:

(x + b / 2a)² = - (c / a) + b² / 4a²

We are about to arrive at our formula!

Step 6: Add - (c / a) and b² / 4a². This gives:

(x + b / 2a)² = (- 4ac + b²) / 4a²

Here we take the LCM (least common multiple) of a and 4a² which is just 4a². Hence, we multiply the numerator and denominator of the constant term by 4a which gave us - 4ac / 4a². We then simplified the RHS as (- 4ac + b²) / 4a².

Step 7: Take the square root of both sides of the equation. This gives us:

x + b / 2a = √{(b² - 4ac) / 4a²}

⇒ x + b / 2a = √(b² - 4ac) / 2a

Here we took the square root in order to simplify the LHS from x² to just x.

Step 8: Rearrange the equation (Write the equation in terms of x). This gives:

x = [√(b² - 4ac) / 2a] - [b / 2a]

Simplifying gives:

x = √(b² - 4ac) - b / 2a

∴ x = [-b ± √(b² - 4ac)] / 2a  (Proved)

There you go! This is the quadratic formula! Now before we end, you might have a question as to why there is a ± (plus-minus) sign before √(b² - 4ac). Well, the square root of any number can be positive or negative.

Let's take an example:

6² is 36

-6² is also 36.

Hence, √36 can be either 6 or -6. Hence we collectively write this as ±6.

But then, you may say that we didn't take √(x + b / 2a)² as ±, we directly wrote it positive. Well, let's clear this with an example.

What is √(6)²? Simply 6.

What is √(-6)²? Simply -6.

How? Well, we are just cancelling of the powers here! The symbol √ means the number is raised to the power 1/2. Hence what we basically are doing is:

{(6)²}^½

We know that when a base has 2 exponents, then they are multiplied. Hence 2 and ½ cancel each other off and we simply get 6.

Applying the same logic to √(x + b / 2a)², here also the powers get cancelled and what we are left with is simply x + b / 2a which we used to derive the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

With this, we have proved the marvelous quadratic formula which helps in everyday life to solve quadratic equations. I hope the explanations gave you a much better understanding of the formula.