What are commonly used formulas used in problem solving?
1 Answer
A few examples...
Explanation:
I will assume that you mean things like common identities and the quadratic formula. Here are just a few:
Difference of squares identity
a^2-b^2 = (a-b)(a+b)
Deceptively simple, but massively useful.
For example:
a^4+b^4 = (a^2+b^2)^2 - 2a^2b^2
color(white)(a^4+b^4) = (a^2+b^2)^2 - (sqrt(2)ab)^2
color(white)(a^4+b^4) = ((a^2+b^2) - sqrt(2)ab)((a^2+b^2) +sqrt(2)ab)
color(white)(a^4+b^4) = (a^2-sqrt(2)ab+b^2)(a^2+sqrt(2)ab+b^2)
Difference of cubes identity
a^3-b^3 = (a-b)(a^2+ab+b^2)
Sum of cubes identity
a^3+b^3 = (a+b)(a^2-ab+b^2)
Quadratic formula
Very useful to know, better if you know how to derive it:
The zeros of
x = (-b+-sqrt(b^2-4ac))/(2a)
Pythagoras theorem
If a right angled triangle has legs of length
c^2 = a^2+b^2
This is also very useful in trigonometric form. If we have an angle
Then:
"hypotenuse"^2 = "adjacent"^2 + "opposite"^2
Dividing both sides by
1 = ("adjacent"/"hypotenuse")^2 + ("opposite"/"hypotenuse")^2
That is:
1 = cos^2 theta + sin^2 theta
Then dividing both sides by
sec^2 theta = 1 + tan^2 theta
Binomial theorem
(a+b)^n = sum_(k=0)^n ((n), (k)) a^(n-k) b^k
where
For example:
(x+1)^4 = x^4+4x^3+6x^2+4x+1