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



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

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

cornitodisc
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yes,,you're right

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

hartnn
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yup, tried that, much interesting ...

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

cornitodisc
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yes,,that's right

apple_pi
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What if we found pq?
(pq)^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^24ac)/a^2
therefore pq = √(b^24ac) /a

apple_pi
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sorry, not √ but ±√

hartnn
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did the same way, good,go on.

apple_pi
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(p+q)+(pq)=2p=2x (because a root is a solution of x)
2x = b/a + √(b^24ac) /a
x = b±√(b^24ac)

2a

apple_pi
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And viola, the quadratic formula

hartnn
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Good Work !

apple_pi
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Thank you

ganeshie8
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Excellent ! somehow ive never seen this before. thank you !!

apple_pi
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Your welcome

lgbasallote
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not satisfied with the completing the square solution huh?

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