Advanced Proof of Quadratic Formula

The Three Important Identities

In this post you wanna dive in and find resourful identities that you thought little of in developnment of quadratic formula.

The general formula of quadratic equation is given by

To get more explanation about the parameters find insightful peek in this basic proof of quadratic formula

The algebraic identities you wanna learn here

1.
2.
3.

Now given the three identities, let’s see how to combine them to produce a path towards the quadratic formula. By dividing through the equation, it inevitabley becomes

Suppose the roots of a quadratic equation are give by r1 and r2 and that they are building blocks towards quadratic formula advancement. Now, form a quadratic equation using the roots. Notice that a is divided through the general quadratic equation above so as to make roots the same

It is crystal clear that both equations have their first term a perfect square-x squared. And that both of them equals to zero. Since they are equivalent to zero and their first term same, let’s equate them.

By observation, it is established that the sum of the roots of the quadratic equation is

And the product is

Step 1

In a special case when the first two identities (1. & 2.) are combined, then

The relationship of the combined identities becomes

Express the left side linearly by taking square roots both sides

Substitute the sum and product of the general equation into the combined identity . It relates as

Step 2

By rearranging, make 4r1r2 the subject of the formula

Let’s use the difference of two squares to find r1 from this equation. Divide both sides by 4r2 , so r1 becomes

The difference of two squares property

Exploit the difference of two squares

Now, after applying difference of two squares, r1 simplifies to

Plug in the parameters given that r1+r2 and r1-r2 are known

When

it reduces to

and we get

When

the other root is

And by combining the solutions we obtain

Proof quadratic formula by converting first term into a perfect square

Now, given the general quadratic equation, you are required to make the first term a perfect square.

Converting the first term into a perfect square is essential so that the quadratic equation resembles the form of this identity

Multiply the equation by a to boths sides

The equation becomes

Rewrite and make the coefficients explict as in general form of quadratic equation

It is time now to complete the square. Take half the coeffient of ax, square the result and add to both sides of the equation or add and subtract the result if all the parameters are on the same side in this case LHS. But you can choose to keep all elements in one side of the equation at initial and intermediate stages as I have done. The coefficient of ax is b . Let’s quickily complete the square

Transform the left side into a perfect square

Find the LCM on the right side and simplify the fraction

Take square roots on both sides to transform the equation into linear

It reduces to

Take all parameters to the right side to make x the subject of the formula

Divide both sides by a to find x as

Leave a Reply