Quadratic equations.
We have all learnt about them in school and not all of us enjoyed solving them.
Partially because of the tidious formula invented 1400 years ago!
Which is (-b±√b²-4ac)/2a for the standard equation ax² + bx + c = 0 (a≠0)
Not everyone may be able to remember this formula.
So for those of you who cant! here is a new and simple way-
ax² + bx + c = 0 (a≠0)
Divide both sides a
x² + bx/a + c/a = 0
or
x² + Bx + C = 0 (B and C are constants)
lets take 2 variables m and n as the roots
The average of these two roots should be equal to -B/2.
The product of these two roots should be equal to C.
To get the average of -B/2, we can have m equal to -B/2 + z and n equal to -B/2 - z
To get the product C, mn = C or (-B/2 + z)(-B/2 - z)
C = B²/4 - z²
z² = B²/4 - C
z = ±√(B²/4 - C)
x = -B/2 ± √(B²/4 - C)
By doing this, we have actually found the two roots m and n.
This method is easier because the trick is to not memorize anything, all you need to do is consider values for the roots = -B/2 ± z
Setting the product equal to C, and solving for z.
Example:
3x² + 18x + 36 = 0
dividing by a,
x² - 6x + 8 = 0
Average is 3. (-B/2)
m = (3+z), n = (3-z)
(3+z)(3-z) = 8 . (C)
9-z² = 8
z² = 9-8
z = ±1
m = 3+z = 4
n = 3-z = 2
Hence 2,4 are the roots.
This method was put together by Po-Shen Loh,
He has done his PhD in mathematics from the Princeton University and is currently the coach for the American IMO team.
Comments