# Step-by-Step Solutions

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Physics Formulas

do physics calculations step by step

work W=30J, d=100m
photon energy 435nm
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Chemical Structures

get a step-by-step procedure to draw the Lewis structures of molecules

What is the Lewis structure of NO2
Lewis dot structure of ATP
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Chemical Conversions and Properties

follow the steps for unit conversion

convert 5mL benzene to moles
how many molecules are in 10 grams of ATP
compute oxidation states of chemicals

oxidation states of chromium(III) chloride
elemental oxidation state of copper(II) sulfate
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Chemical Reactions

learn how to balance a chemical reaction

glucose + oxygen -> water + carbon dioxide
Cu + HNO3 -> Cu(NO3)2 + NO2 + H2O
calculate reaction stoichiometry

0.2 mol CH4 + O2 -> 7 mL H2O + CO2
benzene + nitronium (ion) -> 0.02 mols nitrobenzene + hydrogen (ion)
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Arithmetic

check your work using step-by-step arithmetic

60431 / 89
9 (3+1) + 17 / (6-12)
follow the steps to convert an improper fraction to a mixed number

87/5 as a mixed number
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Number Theory

find step-by-step solutions for prime factorization, primality testing, GCD, and more

is 73 prime?
lcm(90, 342)
are 12 and 21 coprime?
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Statistics

perform basic statistical analysis on a set of data, seeing the steps along the way

average of {1, 2, 3, 4, 5, 6, 7}
mode of {1, 5, 4, 2, 3, 4, 5, 11, 4, 11, 20}
variance 4, 6, 7, 9, 10, 11
quartile of {1, 2, 3, 4, 5, 6}
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Intercepts

find axis intercepts one step at a time

x intercepts of sin(x)
y intercept of x^2+1
intercepts of (x^2+2)(x^2-1)
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Trigonometry

see the steps toward proving a trigonometric identity

(1 + tan(x))/(1 – tan(x)) = (cos(x) + sin(x))/(cos(x) – sin(x))
cot(t/2)^2 = (1 + cos(t)) / (1 – cos(t))
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Algebra

evaluate an expression at a point one step at a time

evaluate x^2 + 2(x + 7) - 19 at x = 7
(x - 1)(x + 3)^2 + 7 where x = 2
solve equations one step at a time

x^2 + 5x + 6 = 0
solve (9^(x + 1)) – (28 (3^(x))) + 3 = 0 over the real numbers
factor polynomials step by step

factor x^4+4
factor x^3+2x^2+x+2
expand polynomials using FOIL, the binomial theorem, and other methods

expanded form of (x + 3)^2
multiply out (x^2 + x)^4
choose from multiple methods to complete the square

complete the square x^2 + 2x + 3
complete the square x^2 + y^2 + 2x + 4y - 11 = 0
simplify rational expressions with steps provided

simplify (1/a + 1/b) (a + b)
k/(k + 1) + 1/((k + 1) (k + 2))
learn to rewrite a rational function using our step-by-step partial fraction decomposition

partial fraction decomposition 1/(x^2 + 4x + 3)
1/(x^3 + 4x^2 + 5x + 2)
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Calculus

find the derivative using the product rule, chain rule, and other methods

derivative of x^4 + 9x^3 + 7x – 2
slope of log(x) / (x^2 + 1)
calculate an integral by substitution, integration by parts, and other methods

integrate sin(x)cos(x)^2
integral of sqrt(a^2 – x^2)
show the integration: arcsec(sqrt(t))
learn how to take a limit

limit of (x – 3) / (x^2 – 2x – 3) as x approaches 3
take the limit as x goes to infinity: (1 + 1/x)^x
find local and global extrema using a variety of tests

local extrema of x^3 - 3x + 1
find the extrema of 3x^4 - 4x^3 - 12x^2 + 1
maximize sin(x)
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Discontinuities

get hints for discontinuities

where is (x^2 - 4)/(x^2 - 2x - 3) discontinuous?
discontinuities of sin((x^2-x)/(x-1)) + log(sqrt(abs(x)))
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Differential Equations

see how ordinary differential equations are solved

y‘(t) – 2y(t) = 3 e^(2t)
t^2 y‘(t) + 2t y(t) = t^4 y(t)^2 + 4
y''+ y = sin(2x)
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Linear Algebra

write a matrix in reduced row echelon form one step at a time

{{1,1,5},{1,-1,1}} row reduce
reduced row echelon form: {{1, -3, 3, -4}, {2, 3, -1, 15}, {4, -3, -1, 19}}
find the determinant step by step with various methods

determinant of {{1,2}, {-1, 2}}
determinant of {{1,2,1}, {1,1,0}, {0,1,1}}
invert a matrix one step at a time

inverse {{1, 1, 2}, {-1, 2, 2}, {3, 2, 3}}
compute a cross product step by step

(1,2,3)x(3,4,5)
compute eigenvalues and eigenvectors step by step

eigenvalues {{3,-1},{0,2}}
eigenvectors {{7,0,-3},{-9,-2,3},{18,0,-8}}
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Mathematical Induction

prove a sum identity by induction

prove by induction sum of j from 1 to n = n(n+1)/2 for n>0
prove divisibility by induction

using induction, prove 9^n-1 is divisible by 4 assuming n>0
prove an inequality by induction

prove by induction (3n)! > 3^n (n!)^3 for n>0