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Derivation of formulae for quadratic equation... - Class 10 PDF Download

A Quadratic Equation looks like this:

Derivation of formulae for quadratic equation... - Class 10

And it can be solved using the Quadratic Formula:

Derivation of formulae for quadratic equation... - Class 10

That formula looks like magic, but you can follow the steps to see how it comes about.

1. Complete the Square

ax2  bx c has "x" in it twice, which is hard to solve.

But there is a way to rearrange it so that "x" only appears once. It is called Completing the Square (please read that first!).

Our aim is to get something like x2  2dx d2, which can then be simplified to (x d)2

So, let's go:

Start withDerivation of formulae for quadratic equation... - Class 10
Divide the equation by aDerivation of formulae for quadratic equation... - Class 10
Put c/a on other sideDerivation of formulae for quadratic equation... - Class 10
Add (b/2a)2 to both sidesDerivation of formulae for quadratic equation... - Class 10

The left hand side is now in the x2  2dx d2format, where "d" is "b/2a"
So we can re-write it this way:

"Complete the Square"Derivation of formulae for quadratic equation... - Class 10

Now x only appears once and we are making progress.

2. Now Solve For "x"

Now we just need to rearrange the equation to leave "x" on the left 

Start withDerivation of formulae for quadratic equation... - Class 10
Square rootDerivation of formulae for quadratic equation... - Class 10
Move b/2a to rightDerivation of formulae for quadratic equation... - Class 10

That is actually solved! But let's simplify it a bit:

Multiply right by 2a/2aDerivation of formulae for quadratic equation... - Class 10
Simplify:Derivation of formulae for quadratic equation... - Class 10


Which is the Quadratic formula we all know and love:

Derivation of formulae for quadratic equation... - Class 10

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