# How to calculate effectively (mathematics)

Moderator
Often you can speed up your calculations a lot my relying more on your memory and using the most effective formulas.

Code:
``````equation    solutions
x²+2DX+E=0  x=-D±√(D²-E)
ax²+bx+c=0  x=(-b±√(b²-4ac))/(2a)``````
You need to properly remember these formulas to do well on math tests (saving time) it is however not particularly useful outside writing exams.

Moderator
quick polynomial division
You can factor polynomials quickly and easily

In the video they make these notices below but you can remember that in your head easily.

Moderator
Square root approximation

This approximation can be approved by adding a second correction term

Code:
``√(x+a) = √x+a/(2√x)-a^2/8/x^(-3/2)+...``
Then instead of overestimating the result it will underestimate it slightly. Instead of dividing by 8 we can divide by 10 and then it will always overestimate the result unless the perfect square is smaller than 5.

u = root of perfect square

a = distance

root = u + a/(2u)-a²/(10u³)

√105 10 + 5/20 - 25/(10·10³) = 10.25-0.0025 = 10.2475

Error = +0.00055

For small square roots you can add 2 decimal places

√2.00 ≈ 1.40 + 0.04/2.8 ≈ 1.4142 (rounding down to compensate for error in the approximation)

Or you can just remember the exact decimals

√2 = 1.41421356265623730950488016887242096980785696718753769 and so on