How to calculate effectively (mathematics)

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

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.

 

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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.

 

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Square root approximation

This approximation can be approved by adding a second correction term

√(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