Question

NOT A QUESTION (JUST INTERESTING)
Alternate derivation of the quadratic formula

Answer 1

You may be familiar with the derivation of the quadratic formula by completing the square...

Answer 2

We can get the exact same result by using the sum and products of roots

Answer 3

yes,,you're right

Answer 4

given a quadratic equation: ax^2+bx+c=0
let the two roots be p and q (which are equal to x)

Answer 5

yup, tried that, much interesting ...

Answer 6

p+q=-b/c
pq=c/a

Answer 7

yes,,that's right

Answer 8

What if we found p-q?

(p-q)^2 = p^2 - 2pq + q^2
= (p^2 + q^2) - 2pq
= (p+q)^2 - 2pq - 2pq
= (p+q)^2 - 4pq
= b^2/a^2 - 4*c/a
= (b^2-4ac)/a^2
therefore p-q = √(b^2-4ac) /a

Answer 9

sorry, not √ but ±√

Answer 10

did the same way, good,go on.

Answer 11

(p+q)+(p-q)=2p=2x (because a root is a solution of x)
2x = -b/a + √(b^2-4ac) /a
x = -b±√(b^2-4ac)
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2a

Answer 12

And viola, the quadratic formula

Answer 13

Good Work !

Answer 14

Thank you

Answer 15

Excellent ! somehow ive never seen this before. thank you !!

Answer 16

Your welcome

Answer 17

not satisfied with the completing the square solution huh?

Answer 18

nah, long at messy.. this way seems so much more elegant