A couple months ago, I was challenged to discover a new way to solve quadratic equations with one rule: I must think of it myself. At first I was hopeless, trying various sorts of random things, but I finally found one way. My way is somewhat limited (extremely disappointing), however, but it still works in most cases. You must only know two things about the quadratic equation:
- The leading coefficient of x^2 is 1
- The difference of the two possible values of x
For example, you have a randomly-picked polynomial with the requirements stated above:
Psst! The answer is:
Which is derived from:
Obviously, this can be solved in the matter of a second by simple factoring, but there is another way:
let y = greater value of x (-2) let z = lesser value of x (-3)
Now, you have two new variables, y and z, but how do you use them? Make new equations!
Somehow, I pulled two magic equations out of thin air:
y^2-(difference*y)=coefficient c z^2+(difference*z)=coefficient c
So, for our current example, the two equations would look like this:
Yay! Now we have a system of equations with like variables! The rest can be solved with simple linear combination.
y^2-1y=6 and y^2+5y+6=0 (y^2+5y=-6)
Adding the two equations gives:
Divide the whole equation by 2:
Subtract 2y from both sides of the equation:
Divide both sides by y:
Yay! That is the correct answer from above! Now, let's start the same super long process for z(I totally don't want to do this)!
The two equations for z again so you don't have to look up there again(I know, I'm a nice person):
z^2+1z=6 and z^2+5z+6 (z^2+5z=-6)
Adding the two equations together gives:
Divide the whole equation by 2 gives:
Subtracting 3z from both sides of the equation gives:
Dividing both sides of the equation by z gives:
Yay! The other solution for x! Now we have the same solution set as we had by factoring:
Phew! Now that my past hour has been used writing this post, I am so tired! You guys are probably really tired too from reading a post from a crazy middle-schooler! Can you please try to help me expand my idea? Help will include:
1. Constructive Criticism 2. Deriving a Formula to Speed up this process 3. Find more ways to use this in quadratic equation solving
Thanks so much!
P.S. I didn't actually have to write out every step. I just wrote it for clarification for the readers.