total maths formulae

8/8/2019 Total Maths Formulae

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MATHEMATICAL FORMULAE

Algebra

1. (a + b)2 = a2 + 2ab + b2; a2 + b2 = (a + b)2

−2ab

2. (a − b)2 = a2 − 2ab + b2; a2 + b2 = (a − b)2 + 2ab

3. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

4. (a + b)3 = a3 + b3 + 3ab(a + b); a3 + b3 = (a + b)3 − 3ab(a + b)

5. (a − b)3 = a3 − b3 − 3ab(a − b); a3 − b3 = (a − b)3 + 3ab(a − b)

6. a2 − b2 = (a + b)(a − b)

7. a3 − b3 = (a − b)(a2 + ab + b2)

8. a3 + b3 = (a + b)(a2 − ab + b2)

9. an

−bn = (a

−b)(an−1 + an−2b + an−3b2 +

· · ·+ bn−1)

10. an = a.a.a . . . n times

11. am.an = am+n

12.am

an= am−n if m > n

= 1 if m = n

=1

an−mif m < n; a ∈ R, a = 0

13. (am)n = amn = (an)m

14. (ab)n = an.bn

15.

ab

n= a

n

bn

16. a0 = 1 where a ∈ R, a = 0

17. a−n =1

an, an =

1

a−n

18. a p/q = q√

a p

19. If am = an and a = ±1, a = 0 then m = n

20. If an = bn where n = 0, then a = ±b

21. If √

x,√

y are quadratic surds and if a +√

x =√

y, then a = 0 and x = y

22. If √

x,√

y are quadratic surds and if a +√

x = b +√

y then a = b and x = y

23. If a,m,n are positive real numbers and a = 1, then loga mn = loga m+loga n

24. If a,m,n are positive real numbers, a = 1, then loga

m

n

= loga m− loga n

25. If a and m are positive real numbers, a = 1 then loga mn = n loga m

26. If a, b and k are positive real numbers, b = 1, k = 1, then logb a =logk a

logk b

27. logb a =1

loga bwhere a, b are positive real numbers, a = 1, b = 1

28. if a,m,n are positive real numbers, a = 1 and if loga m = loga n, then

m = n

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2

29. if a + ib = 0 where i =√−1, then a = b = 0

30. if a + ib = x + iy, where i =√−1, then a = x and b = y

31. The roots of the quadratic equation ax2+bx+c = 0; a = 0 are−b ±√

b2 − 4ac

2a

The solution set of the equation is−b + √∆

2a, −b −√∆

2a

where ∆ = discriminant = b2 − 4ac

32. The roots are real and distinct if ∆ > 0.

33. The roots are real and coincident if ∆ = 0.

34. The roots are non-real if ∆ < 0.

35. If α and β are the roots of the equation ax2 + bx + c = 0, a = 0 then

i) α + β =−b

a= − coeff. of x

coeff. of x2

ii) α · β = ca

= constant termcoeff. of x2

36. The quadratic equation whose roots are α and β is (x − α)(x − β ) = 0

i.e. x2 − (α + β )x + αβ = 0

i.e. x2 − Sx + P = 0 where S =Sum of the roots and P =Product of the

roots.

37. For an arithmetic progression (A.P.) whose first term is (a) and the common

difference is (d).

i) nth term= tn = a + (n

−1)d

ii) The sum of the first (n) terms = S n = n2

(a + l) = n2{2a + (n − 1)d}

where l =last term= a + (n − 1)d.

38. For a geometric progression (G.P.) whose first term is (a) and common ratio

is (γ ),

i) nth term= tn = aγ n−1.

ii) The sum of the first (n) terms:

S n =a(1 − γ n)

1

−γ

if γ < 1

= a(γ n − 1)γ − 1

if γ > 1

= na if γ = 1

.

39. For any sequence {tn}, S n − S n−1 = tn where S n =Sum of the first (n)

terms.

40.n

γ=1γ = 1 + 2 + 3 + · · · + n =

n

2(n + 1).

41.n

γ=1γ 2 = 12 + 22 + 32 + · · · + n2 =

n

6(n + 1)(2n + 1).



3

42.n

γ=1γ 3 = 13 + 23 + 33 + 43 + · · · + n3 =

n2

4(n + 1)2.

43. n! = (1).(2).(3). . . . .(n − 1).n.

44. n! = n(n − 1)! = n(n − 1)(n − 2)! = . . . . .

45. 0! = 1.46. (a + b)n = an + nan−1b +

n(n − 1)

2!an−2b2 +

n(n − 1)(n − 2)

3!an−3b3 + · · ·+

bn, n > 1.



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Mensuration:

1. Area of a circle of radius (γ ) is πγ 2.

2. Perimeter (or circumference) of a circle of radius (γ ) is 2πγ .

3. Volume of a sphere of radius (γ ) is 43πγ 3.

4. Surface area of a sphere of radius (γ ) is 4πγ 2.

5. Volume of right circular cone is1

3πγ 2h where (γ ) is the radius of the base

and (h) is the height of the cone.

6. Curved surface area of a cone is πγl where (l) is the slant height of the cone.

7. Total surface area of a cone is πγl + πγ 2.

8. Volume of a right circular cylinder is πγ 2h.

9. Curved surface area of a right circular cylinder is 2πγh.

10. Total surface area of a right circular cylinder is 2πγh + 2πγ 2.

11. Area of a triangle=1

2(base) (height).

12. Area of a rectangle = (length) (breadth).

13. Perimeter of a rectangle= 2 (length) +2 (breadth).

14. Area of a square = (Side)2.

15. Perimeter of a square = 4 (Side).

16. Volume of a cube= x3

where (x) is the length of the side.

17. Surface area of a cube= 6x2.

18. Area of a trapezium=1

2(sum of parallel side) × (distance between the

parallel sides).

19. Area of an equilateral triangle =

√3

4a2 =

1√3 p2.

where (a) is the length of a side and ( p) is the length of an altitude.

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2

Derivatives:

1. f (x) = limh→0

f (x + h)− f (x)

hprovided the limit exists.

2. f (a) = limh→0

f (a + h)− f (a)h

provided the limit exists.

3.d

dx(k) = 0 where (k) is a constant.

4.d

dx(u + v) =

du

dx+

dv

dx,

5.d

dx(u− v) =

du

dx− dv

dx.

6.d

dx(uv) = u

dv

dx+ v

du

dx.

7.d

dx

uv

=

v dudx− u dv

dx

v2(v = 0).



Differentiation Formulas

d

dxk = 0 (1)

d

dx [f (x) ± g(x)] = f

(x) ± g

(x) (2)

d

dx[k · f (x)] = k · f (x) (3)

d

dx[f (x)g(x)] = f (x)g(x) + g(x)f (x) (4)

d

dx

f (x)

g(x)

=

g(x)f (x) − f (x)g(x)

[g(x)]2(5)

d

dxf (g(x)) = f (g(x)) · g(x) (6)

d

dx x

n

= nx

n−1

(7)d

dxsinx = cosx (8)

d

dxcosx = − sinx (9)

d

dxtanx = sec2 x (10)

d

dxcotx = − csc2 x (11)

d

dxsecx = secx tanx (12)

d

dxcscx = − cscx cotx (13)

d

dxex = ex (14)

d

dxax = ax lna (15)

d

dxln |x| =

1

x(16)

d

dxsin−1 x =

1√1 − x2

(17)

ddx

cos−1 x = −1√1 − x2

(18)

d

dxtan−1 x =

1

x2 + 1(19)

d

dxcot−1 x =

−1

x2 + 1(20)

d

dxsec−1 x =

1

|x|√x2 − 1(21)

d

dxcsc−1 x =

−1

|x|√x2 − 1(22)

Integration Formulas

dx = x + C (1)

xn dx = x

n+1

n + 1+ C (2)

dx

x= ln |x| + C (3)

ex dx = ex + C (4)

ax dx =

1

lnaax + C (5)

lnxdx = x lnx

−x + C (6)

sinxdx = − cosx + C (7)

cosx dx = sinx + C (8)

tanx dx = − ln | cosx| + C (9)

cotx dx = ln | sinx| + C (10)

secxdx = ln | secx + tanx| + C (11)

cscxdx = − ln | cscx + cotx| + C (12)

sec2 xdx = tanx + C (13)

csc2 xdx = − cotx + C (14)

secx tanx dx = secx + C (15)

cscx cotxdx = − cscx + C (16)

dx√

a2 − x2= sin−1

x

a+ C (17)

dx

a2 + x2=

1

atan−1

x

a+ C (18)

dx

x√x2 − a2

=1

asec−1

|x|a

+ C (19)



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Formulas from Calculus

Derivativesddx

[xn] = nxn−1 ddx

[ex] = ex ddx

[sin x] = cos x

ddx

[c] = 0 ddx

[bx] = bx ln b ddx

[cos x] = − sin x

ddx

[x] = 1 ddx

[ln x] = 1x

ddx

[tan x] = sec2 x

ddx

1x

= − 1

x2ddx

[logb x] = 1x ln b

ddx

[sec x] = tan x sec x

ddx

1x2

= − 2

x3ddx

[sinh x] = cosh x ddx

[arcsin x] = 1√1−x2

ddx

[√

x] = 12√x

ddx

[cosh x] = sinh x ddx

[arctan x] = 11+x2

ddx

1√x

= − 1

2x√x

ddx

[tanh x] = sech 2x

ddx

[arcsinh x] = 1√1+x2

ddx [arctanh x] = 11−x2

Product Rule: ddx

[f (x)g(x)] = f (x)g(x) + f (x)g(x)

Quotient Rule:d

dx

f (x)

g(x)

=

g(x)f (x) − f (x)g(x)

g(x)2

Chain Rule: ddx

[f (g(x))] = f (g(x))g(x)

Special Cases

d

dx[f (x)n] = nf (x)n−1f (x)

d

dx

1

g(x)

=

−g(x)

g(x)2

d

dx[ln |f (x)|] =

f (x)

f (x)

d

dx

ef (x)

= f (x)ef (x)



Integrals

xn dx = 1

n+1xn+1 + C

1x

dx = ln |x| + C c dx = cx + C x dx = 1

2x2 + C x2 dx = 1

3x3 + C 1

x2dx = −1

x+ C √

x dx = 23x√

x + C

1√

xdx = 2√x + C

1

1 + x2dx = arctan x + C

1√1 − x2

dx = arcsin x + C ln x dx = x ln x − x + C xn ln x dx = xn+1

n+1 ln x − xn+1

(n+1)2+ C

ex dx = ex + C

bx dx = 1

ln bbx

sinh x dx = cosh x + C cosh x dx = sinh x + C sin x dx = − cos x + C cos x dx = sin x + C tan x dx = ln | sec x| + C

sec x dx = ln | tan x + sec x| + C sin2 x dx = 1

2(x − sin x cos x) + C cos2 x dx = 1

2(x + sin x cos x) + C tan2 x dx = tan x − x + C sec2 x dx = tan x + C

Substitution

f (g(x))g(x) dx =

f (u) du = F (u) + C = F (g(x)) + C

ba

f (g(x))g(x) dx =

g(b)g(a)

f (u) du

Special cases

f (x)

f (x)dx = ln |f (x)| + C

ef (x)f (x) dx = ef (x) + C

By parts

u dv = uv −

v du

ba

u dv = uv]ba − ba

v du

or

f (x)g(x)dx = f (x)g(x) −

g(x)f (x) dx

ba

f (x)g(x)dx = f (x)g(x)]ba − ba

g(x)f (x) dx



Co-ordinate Geometry

1. Distance between two points P (x1, y1) and Q(x2, y2) is given by:

d(P,Q) =

(x2 − x1)2 + (y2 − y1)2 {Distance formula}

2. Distance of a pointP (x, y) from the origin is given by d(0, P ) = x2 + y2.

3. Equation of the x-axis is y = 0

4. Equation of the y-axis is x = 0

5. Equation of a straight line parallel to x-axis and passing through the point

P (a, b) is y = b.

6. Equation of a straight line parallel to y-axis and passing through the point

P (a, b) is x = a.

7. Slope of a straight line= m = tan θ = y2

− y1

x2 − x1where (θ) is the inclination

of the straight line and (x1, y1) and (x2, y2) are any two points on the line.

8. Equation of a line in the slope-intercept form is y = mx + b.

9. Equation of a straight line in point-slope form is y − y1 = m(x− x1).

10. Equation of a straight line in two-points form isy − y1

y2 − y1=

x− x1

x2 − x1.

11. Equation of a straight line in double-intercept form is:x

a+

y

b= 1.

12. For a straight line whose equation is ax + by + c = 0

i) slope= −ab

ii) y-intercept= −cb

iii) x- intercept= − ca

.

13. The straight lines with slopes (m) and (m) are mutually perpendicular if

m,m = −1.

14. The straight line with slopes (m) and (m) are parallel to each other if

m = m.

Typeset by AMS -TEX



© 2005 Paul Dawkins

Trig Cheat Sheet

Definition of the Trig Functions

Right triangle definition For this definition we assume that

02

π θ < < or 0 90θ ° < < ° .

oppositesinhypotenuse

θ = hypotenusecscopposite

θ =

adjacentcos

hypotenuseθ =

hypotenusesec

adjacentθ =

oppositetan

adjacentθ =

adjacentcot

oppositeθ =

Unit circle definition

For this definition θ is any angle.

sin1

y yθ = =

1csc

yθ =

cos1

x xθ = =

1sec

xθ =

tany

xθ = cot

x

yθ =

Facts and PropertiesDomain

The domain is all the values of θ that

can be plugged into the function.

sinθ , θ can be any angle

cosθ , θ can be any angle

tanθ ,1

, 0, 1, 2,2

n nθ π ⎛ ⎞

≠ + = ± ±⎜ ⎟⎝ ⎠

…

cscθ , , 0, 1, 2,n nθ π ≠ = ± ± …

secθ ,1

, 0, 1, 2,2

n nθ π ⎛ ⎞

≠ + = ± ±⎜ ⎟⎝ ⎠

…

cotθ , , 0, 1, 2,n nθ π ≠ = ± ± …

Range The range is all possible values to getout of the function.

1 sin 1θ − ≤ ≤ csc 1 and csc 1θ θ ≥ ≤ −

1 cos 1θ − ≤ ≤ sec 1 and sec 1θ θ ≥ ≤ −

tanθ −∞ ≤ ≤ ∞ cotθ −∞ ≤ ≤ ∞

Period

The period of a function is the number,

T , such that ( ) ( ) f T f θ θ + = . So, if ω

is a fixed number and θ is any angle we

have the following periods.

( )sin ωθ → 2

T π

ω =

( )cos ωθ → 2

T π

ω =

( )tan ωθ → T

π

ω =

( )csc ωθ → 2

T π

ω =

( )sec ωθ → 2

T π

ω =

( )cot ωθ → T π

ω =

θ

adjacent

opposite

hypotenuse

x

y

, x

θ

x

y1




Formulas and IdentitiesTangent and Cotangent Identities

sin costan cot

cos sin

θ θ θ θ

θ θ = =

Reciprocal Identities

1 1csc sinsin csc

1 1sec cos

cos sec

1 1cot tan

tan cot

θ θ θ θ

θ θ θ θ

θ θ θ θ

= =

= =

= =

Pythagorean Identities 2 2

2 2

2 2

sin cos 1

tan 1 sec

1 cot csc

θ θ

θ θ

θ θ

+ =

+ =

+ =

Even/Odd Formulas

( ) ( )

( ) ( )

( ) ( )

sin sin csc csc

cos cos sec sec

tan tan cot cot

θ θ θ θ

θ θ θ θ

θ θ θ θ

− = − − = −

− = − =

− = − − = −

Periodic Formulas

If n is an integer.

( ) ( )

( ) ( )( ) ( )

sin 2 sin csc 2 csc

cos 2 cos sec 2 sectan tan cot cot

n n

n nn n

θ π θ θ π θ

θ π θ θ π θ θ π θ θ π θ

+ = + =

+ = + =+ = + =

Double Angle Formulas

( )

( )

( )

2 2

2

2

2

sin 2 2sin cos

cos 2 cos sin

2cos 1

1 2sin

2tantan 2

1 tan

θ θ θ

θ θ θ

θ

θ

θ θ

θ

=

= −

= −

= −

=

−

Degrees to Radians Formulas

If x is an angle in degrees and t is an

angle in radians then

180and

180 180

t x t t x

x

π π

π = ⇒ = =

Half Angle Formulas

( )( )

( )( )( )( )

2

2

2

1sin 1 cos 2

2

1

cos 1 cos 22

1 cos 2tan

1 cos 2

θ θ

θ θ

θ θ

θ

= −

= +−

=+

Sum and Difference Formulas

( )

( )

( )

sin sin cos cos sin

cos cos cos sin sin

tan tantan

1 tan tan

α β α β α β

α β α β α β

α β α β

α β

± = ±

± =

±± =

∓

∓

Product to Sum Formulas

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1sin sin cos cos

2

1cos cos cos cos

2

1sin cos sin sin

2

1cos sin sin sin

2

α β α β α β

α β α β α β

α β α β α β

α β α β α β

= − − +⎡ ⎤⎣ ⎦

= − + +⎡ ⎤⎣ ⎦

= + + −⎡ ⎤⎣ ⎦

= + − −⎡ ⎤⎣ ⎦

Sum to Product Formulas

sin sin 2sin cos2 2

sin sin 2cos sin2 2

cos cos 2cos cos2 2

cos cos 2sin sin2 2

α β α β α β

α β α β α β

α β α β α β

α β α β α β

+ −⎛ ⎞ ⎛ ⎞+ = ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ −⎛ ⎞ ⎛ ⎞− = ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

+ −⎛ ⎞ ⎛ ⎞+ = ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

+ −⎛ ⎞ ⎛ ⎞− = − ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠Cofunction Formulas

sin cos cos sin2 2

csc sec sec csc2 2

tan cot cot tan2 2

π π θ θ θ θ

π π θ θ θ θ

π π θ θ θ θ

⎛ ⎞ ⎛ ⎞− = − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− = − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− = − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠




Unit Circle

For any ordered pair on the unit circle ( ), x y : cos xθ = and sin yθ =

Example

5 1 5 3cos sin

3 2 3 2

π π ⎛ ⎞ ⎛ ⎞= = −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

3

π

4

π

6

π

2 2,

2 2

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

3 1,

2 2

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

1 3,

2 2

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

60°

45°

30°

2

3

π

3

4

π

5

6

π

7

6

π

5

4

π

4

3

π

11

6

π

7

4

π

5

3

π

2

π

π

3

2

π

0

2π

1 3,

2 2

⎛ ⎞−⎜ ⎟

⎝ ⎠

2 2,

2 2

⎛ ⎞−⎜ ⎟

⎝ ⎠

3 1,

2 2

⎛ ⎞−⎜ ⎟

⎝ ⎠

3 1,

2 2

⎛ ⎞− −⎜ ⎟⎝ ⎠

2 2,

2 2

⎛ ⎞− −⎜ ⎟

⎝ ⎠

1 3,

2 2

⎛ ⎞− −⎜ ⎟

⎝ ⎠

3 1,

2 2

⎛ ⎞−⎜ ⎟

⎝ ⎠

2 2,

2 2

⎛ ⎞−⎜ ⎟

⎝ ⎠

1 3,

2 2

⎛ ⎞−⎜ ⎟

⎝ ⎠

( )0,1

( )0, 1−

( )1,0−

90°

120°

135°

150°

180°

210°

225°

240°270° 300°

315°

330°

360°

0°

x

)1 0

y




Inverse Trig Functions

Definition 1

1

1

sin is equivalent to sin

cos is equivalent to cos

tan is equivalent to tan

y x x y

y x x y

y x x y

−

−

−

= =

= =

= =

Domain and Range Function Domain Range

1sin y x−= 1 1 x− ≤ ≤

2 2 y

π π − ≤ ≤

1cos y x−= 1 1 x− ≤ ≤ 0 y π ≤ ≤

1tan y x−= x−∞ < < ∞

2 2 y

π π − < <

Inverse Properties ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )

1 1

1 1

1 1

cos cos cos cos

sin sin sin sin

tan tan tan tan

x x

x x

x x

θ θ

θ θ

θ θ

− −

− −

− −

= =

= =

= =

Alternate Notation 1

1

1

sin arcsin

cos arccos

tan arctan

x x

x x

x x

−

−

−

=

=

=

Law of Sines, Cosines and Tangents

Law of Sines

sin sin sin

a b c

α β γ = =

Law of Cosines 2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

a b c bc

b a c ac

c a b ab

α

β

γ

= + −

= + −= + −

Mollweide’s Formula

( )12

12

cos

sin

a b

c

α β

γ

−+=

Law of Tangents

( )( )

( )( )

( )( )

12

12

12

12

12

12

tan

tan

tan

tan

tan

tan

a b

a b

b c

b c

a c

a c

α β

α β

β γ

β γ

α γ

α γ

−−=

+ +

−−=

+ +

−− =+ +

c a

b

β

γ



¡

¡

¡

¡

¡

¡

¡

¡

¡

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j k l m n n n m l n m n n z { l k m { |



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C

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A

d

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0 ¡ c G U f U 0 ¡ c G d f }

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For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins

Algebra Cheat Sheet

Basic Properties & FactsArithmetic Operations

( )

, 0

b abab ac a b c a

c ca

a a acb

bc bc b

c

a c ad bc a c ad bc

b d bd b d bd

a b b a a b a b

c d d c c c c

a

ab ac ad bb c a

ca bc

d

+ = + =

= =

+ −+ = − =

− − += = +

− −

+ = + ≠ =

Exponent Properties

( )

( )

( ) ( )1

1

0

1

1, 0

1 1

nm

m m

nn m n m n m

m m n

mn nm

n nn

n nn

n n

n n

n n n nn

n

aa a a a

a a

a a a a

a aab a b

b b

a aa a

a b ba a a

b a a

+ −−

−−

−

= = =

= = ≠

= =

= =

= = = =

Properties of Radicals

1

, if is odd

, if is even

nn n n n

nm n nm n

n

n n

n n

a a ab a b

a aa a

b b

a a n

a a n

= =

= =

=

=

Properties of Inequalities If then and

If and 0 then and

If and 0 then and

a b a c b c a c b c

a ba b c ac bcc c

a ba b c ac bc

c c

< + < + − < −

< > < <

< < > >

Properties of Absolute Value

if 0

if 0

a aa

a a

≥= − <

0

Triangle Inequality

a a a

aa

ab a b b b

a b a b

≥ − =

= =

+ ≤ +

Distance Formula

If ( )1 1 1,P x y= and ( )2 2 2,P x y= are two

points the distance between them is

( ) ( ) ( )2 2

1 2 2 1 2 1,d P P x x y y= − + −

Complex Numbers

( ) ( ) ( )

( ) ( ) ( )

( )( ) ( )

( )( )

( )

( )( )

2

2 2

2 2

2

1 1 , 0

Complex Modulus

Complex Conjugate

i i a i a a

a bi c di a c b d i

a bi c di a c b d i

a bi c di ac bd ad bc i

a bi a bi a b

a bi a b

a bi a bi

a bi a bi a bi

= − = − − = ≥

+ + + = + + +

+ − + = − + −

+ + = − + +

+ − = +

+ = +

+ = −

+ + = +

http://tutorial.math.lamar.edu/





Logarithms and Log Properties Definition

log is equivalent to y

b y x x b= =

Example3

5log 125 3 because 5 125= =

Special Logarithms

10

ln log natural log

log log common log

e x x

x x

==

where 2.718281828e = K

Logarithm Properties

( )

( )

log

log 1 log 1 0

log

log log

log log log

log log log

b

b b

x x

b

r

b b

b b b

b b b

b

b x b x

x r x

xy x y

x x y

y

= =

= =

=

= +

= −

The domain of logb x is 0 x >

Factoring and SolvingFactoring Formulas

( ) ( )

( )( )

( ) ( ) ( )

2 2

22 2

22 2

2

22

x a x a x a

x ax a x a

x ax a x a

x a b x ab x a x b

− = + −

+ + = +− + = −

+ + + = + +

( )

( )

( )( )

( ) ( )

33 2 2 3

33 2 2 3

3 3 2 2

3 3 2 2

3 3

3 3

x ax a x a x a

x ax a x a x a

x a x a x ax a

x a x a x ax a

+ + + = +

− + − = −

+ = + − +

− = − + +

( )( )2 2n n n n n n

x a x a x a− = − + If n is odd then,

( ) ( )

( )( )

1 2 1

1 2 2 3 1

n n n n n

n n

n n n n

x a x a x ax a

x a

x a x ax a x a

− − −

− − − −

− = − + + +

+

= + − + − +

L

L

Quadratic Formula

Solve 2 0ax bx c+ + = , 0a ≠ 2 4

2

b b ac

x a

− ± −=

If 2 4 0b ac− > - Two real unequal solns.

If 2 4 0b ac− = - Repeated real solution.

If 2 4 0b ac− < - Two complex solutions.

Square Root Property

If 2 x p= then x p= ±

Absolute Value Equations/Inequalities

If b is a positive number

or

or

p b p b p b

p b b p b

p b p b p b

= ⇒ = − =

< ⇒ − < <

> ⇒ < − >

Completing the Square

Solve 22 6 10 0 x x− − =

(1) Divide by the coefficient of the 2 x

2 3 5 0 x x− − = (2) Move the constant to the other side.

2 3 5 x x− =

(3) Take half the coefficient of x, square

it and add it to both sides2 2

2 3 3 9 293 5 5

2 2 4 4 x x

− + − = + − = + =

(4) Factor the left side2

3 29

2 4 x

− =

(5) Use Square Root Property

3 29 29

2 4 2 x − = ± = ±

(6) Solve for x

3 29

2 2 x = ±






Functions and GraphsConstant Function

( )or y a f x a= =

Graph is a horizontal line passing

through the point ( )0,a .

Line/Linear Function

( )or y mx b f x mx b= + = +

Graph is a line with point ( )0,b and

slope m.

Slope

Slope of the line containing the two

points ( )1 1, x y and ( )2 2, x y is

2 1

2 1

rise

run

y y

m x x

−

= =−

Slope – intercept form

The equation of the line with slope m

and y-intercept ( )0,b is

y mx b= +

Point – Slope form

The equation of the line with slope m

and passing through the point ( )1 1, x y is

( )1 1 y y m x x= + −

Parabola/Quadratic Function

( ) ( ) ( )2 2

y a x h k f x a x h k = − + = − +

The graph is a parabola that opens up if

0a > or down if 0a < and has a vertex

at ( ),h k .


( )2 2 y ax bx c f x ax bx c= + + = + +

The graph is a parabola that opens up if

0a > or down if 0a < and has a vertex

at ,2 2

b b f

a a

− − .


( )2 2 x ay by c g y ay by c= + + = + +

The graph is a parabola that opens right

if 0a > or left if 0a < and has a vertex

at ,2 2

b bg

a a

− − .

Circle

( ) ( )2 2 2 x h y k r − + − =

Graph is a circle with radius r and center

( ),h k .

Ellipse

( ) ( )2 2

2 21

x h y k

a b

− −+ =

Graph is an ellipse with center ( ),h k

with vertices a units right/left from thecenter and vertices b units up/down from

the center.

Hyperbola

( ) ( )2 2

2 21

x h y k

a b

− −− =

Graph is a hyperbola that opens left and

right, has a center at ( ),h k , vertices a

units left/right of center and asymptotes

that pass through center with slopeb

a± .

Hyperbola

( ) ( )2 2

2 21

y k x h

b a

− −− =

Graph is a hyperbola that opens up and

down, has a center at ( ),h k , vertices b units up/down from the center and

asymptotes that pass through center with

slopeb

a± .





Common Algebraic ErrorsError Reason/Correct/Justification/Example

20

0≠ and

22

0≠ Division by zero is undefined!

23 9− ≠ 23 9− = − , ( )

23 9− = Watch parenthesis!

( )32 5 x x≠ ( )

32 2 2 2 6 x x x x x= =

a a a

b c b c≠ +

+

1 1 1 12

2 1 1 1 1= ≠ + =

+

2 3

2 3

1 x x

x x

− −≠ ++

A more complex version of the previous

error.

a bx

a

+1 bx≠ +

1a bx a bx bx

a a a a

+= + = +

Beware of incorrect canceling!

( )1a x ax a− − ≠ − − ( )1a x ax a− − = − +

Make sure you distribute the “-“!( )

2 2 2 x a x a+ ≠ + ( ) ( ) ( )2 2 22 x a x a x a x ax a+ = + + = + +

2 2 x a x a+ ≠ + 2 2 2 25 25 3 4 3 4 3 4 7= = + ≠ + = + =

x a x a+ ≠ + See previous error.

( )n n n

x a x a+ ≠ + and n n n x a x a+ ≠ + More general versions of previous threeerrors.

( ) ( )2 2

2 1 2 2 x x+ ≠ +

( ) ( )2 2 22 1 2 2 1 2 4 2 x x x x x+ = + + = + +

( )2 22 2 4 8 4 x x x+ = + +

Square first then distribute!

( ) ( )2 2

2 2 2 1 x x+ ≠ +

See the previous example. You can not

factor out a constant if there is a power on

the parethesis!

2 2 2 2 x a x a− + ≠ − + ( )1

2 2 2 2 2 x a x a− + = − +

Now see the previous error.

a ab

b c

c

≠

1

1

a

a a c ac

b b b b

c c

= = =

a

acb

c b

≠

1

1

a a

a ab b

cc b c bc

= = =

total maths formulae

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