Quiz
Part I:
1. Expand the formulas:
1. a2 − b2 = (a − b)(a + b)
2. ( a − b ) =a2 − 2ab + b2
2
3. ( a + b ) =a2 + 2ab + b2
2
2. How to solve incomplete quadratic equations:
0 -Take x out of the brackets.
a) ax2 + bx =
x ( ax + b ) =
0 . Then equate each part to 0.
x=0
ax + b =
0 x=
−
b
a
0 - Move c to the other side.
b) ax2 + c =
ax2 = − c
c
x2 = −
a
x =± −
c)
c
a
ax2 = 0 - x is always 0.
0 with discriminant?
3. How to solve complete quadratic equations ax2 + bx + c =
=
D b2 − 4ac
If D<0 then there are no real solutions
If D=0 then x = −
If D>0 then x=
b
2a
−b ± D
2a
4. Viet theorem
Viet theorem can be used in two ways:
ax2 + bx + c =
0
b
x1 + x2 =
−
a
c
x1 ⋅ x2 =
a
OR
In reduced quadratic equation, where a = 1
x2 + px + q =
0
x1 + x2 =
−p
x1 ⋅ x2 =
q
5. Factorization of quadratic equation
ax2 + bx + c= a ( x − x1 ) ( x − x2 ) ,
0
where x1 and x2 are solutions of quadratic equation ax2 + bx + c =
6. Completing the square formulas (2 formulas)
2
2
2
2
b

b
x2 + bx =  x +  −  
2


2
b

b
x2 − bx =  x −  −  
2

2
7. What are factors and multiples? What is the difference between them?
Factors are numbers by which we can divide our values.
Multiples are numbers that can be divided by our values.
Example: 2 is a factor of 10; 10 is a multiple of 5.
8. What are:
a) Integer numbers – Numbers without any fractional part. Can be 0, negative, positive, but
not fractional or decimal.
b) Consecutive numbers – Numbers in specific order.
9. Finish the following formulas:
an + m
a) an ⋅ am =
b)
an
= an − m
am
c)
(a ) = a
n
m
n⋅ m
an ⋅ bn
d) ( a ⋅ b ) =
n
 a
n
an
e)   = n
b
b
f)
a− n =
1
an
g) =
a0 1
h)
i)
j)
n
a ≠0
m
am = a n
a ⋅ b = a⋅b
a
b
=
a
b
10. How to find a part of a number?
Multiply the number by that part.
Example: Find
1
1
of 10. ⋅ 10 =
5
2
2
11. How to find a number, knowing the value of its part?
Knowing the previous rule, build an equation and find x.
Example: Find a number,
x⋅
1
=
5
2
x =5÷
1
of which is 5.
2
1
= 5 ⋅ 2 = 10
2
12. How to find what part of first number is the second number?
Divide the second number by first one. (OF goes to denominator, IS goes to numerator)
Example: What part of 75 is 50?
50 2
=
75 3
13. How to find percent of a number?
Convert the percent into part by dividing it by 100 and then multiply the number by that part
as in paragraph 10.
Example: Find 40% of 60.
40% = 0.4 part.
0.4 x 60 = 24
14. How to find a number, knowing the value of its percent?
Convert the percent into part by dividing it by 100 and then build an equation knowing the
previous paragraph.
Example: Find the number, if its 40% is 24.
x ⋅ 0.4 = 24
x = 24 ÷ 0.4 = 60
15. How to find what percent of first number is the second number?
Divide the second number by the first one as we did in 12th paragraph, but additionally
multiply the answer by 100% to convert that into percent.
Example: What percent of 60 is 24?
24
= 0.4
60
0.4 ⋅ 100 =
40%
16. What is direct and inverse proportion? What is the difference between them and what rules
are used in each of them?
-Direct proportion is when you increase or decrease one value, the other one also increases or
decreases in the same proportion. The rule we use there is cross multiplication. Direct
proportion can be written as ratios.
-Inverse proportion is when you increase or decrease one value, the other one decreases or
increases in the same proportion. The rule we use there is line-by-line product. Inverse
proportion cannot be written as ratios.
17. Simple and compound interest. Write down their formulas and explain the difference
between them
Simple interest = linear function = linear increase or decrease (decay)
Compound interest = exponential function = exponential increase or decrease (decay)

Simple ==
Sn S  1 ±

pn 

100 

Compound ==
Sn S  1 ±

p 

100 
n
18. When does a linear equation have infinite number of solutions?
When x is gone from the equation and we get true equality
Example: 5x – 5= 5x – 5  0 = 0
19. When does a linear equation have one solution?
As long as x is in the equation, we can find one solution.
Example: 5x – 3 = 3x – 5  2x = -2  x = -1
20. When does a linear equation have no solutions?
When x is gone from the equation and we get wrong equality
Example: 5x – 5= 5x – 3  0=2
21. When does a system of equations (simultaneous of equations) have infinite number of
solutions?
c1
a x + b1y =
a1 b1 c1
for  1
= =
a2 b2 c2
c2
a2 x + b2 y =
22. When does a system of equations (simultaneous of equations) have one solution?
c1
a x + b1y =
a1 b1
for  1
≠
a2 b2
a
x
b
y
c2
+
=
 2
2
23. When does a system of equations (simultaneous of equations) have no solutions?
c1
a x + b1y =
a1 b1 c1
for  1
=
≠
a2 b2 c2
c2
a2 x + b2 y =
24. How to solve a module equality (for example x − 2 =
6)
Expand it 2 times with opposite signs of the answer
x −2 =
6
and
−6
x −2 =
then find x.
25. When does a module equality have no solutions?
When the value of a module is equal to a negative number.
26. When does a module equality have one solution?
When the value of a module is equal to 0.
27. Rate problems. What is the formulas for Rate, time and distance?
D= R ⋅ T
D
R=
T
D
T=
R
Average rate =
Total dis tan ce
Total time
Part II:
28. Linear function and its 4 types
=
y mx + c
y = mx
- lines with slope
x = a - vertical line (line perpendicular to x-axis and/or parallel to y –axis)
y = c - horizontal line (line perpendicular to y-axis and/or parallel to x-axis)
Line x=a is parallel to its own type (vertical line) x=a and perpendicular to horizontal line y=c
Line y=c is parallel to its own type (horizontal line) y=c and perpendicular to a vertical line x=a
29. What is slope (all variations) and how it affects the line?
-Slope shows change of y-value for each change in x-value
-Slope shows the steepness of the line.
-Slope shows whether the line is increasing or decreasing. If m>0 the line is increasing. If m<0
the line is decreasing
-Slope shows the average rate of change or just rate of change (change can be increase or
decrease)
30. What is the formula for finding the slope?
y2 − y1
y − y2
or 1
x2 − x1
x1 − x2
31. What is the slope of a vertical and horizontal lines?
Vertical line x=a always has an undefined slope.
Horizontal line y=c always has a slope that is equal to 0.
32. What is the condition for two lines to be parallel?
Slopes are equal, y-intercepts are different.
=
m1 m2 and c1 ≠ c2
33. What is the condition for two lines to just intersect?
Slopes are different. m1 ≠ m2
34. What is the condition for two lines to be perpendicular?
1
or m1 ⋅ m2 =
−1
m2
Slopes are negative reciprocals of one another. m1 =
−
In other words, just flip the slope and change the sign of it to find the slope of a perpendicular
line.
35. How to find midpoint of a segment?
x1 + x2
2
y1 + y2
y0 =
2
x0 =
36. How to find a distance between two points or length of a segment?
d=
( x1 − x2 ) + ( y1 − y2 )
2
2
37. Quadratic function and its 3 types
y = ax2 + bx + c - standard form (y-intercept form)
y = a ( x − x 0 ) + y 0 - vertex form (turning point form)
2
y=
a ( x − x1 ) ( x − x2 ) - x-intercept form ( x1 and x2 are x-intercepts)
38. How to find vertex in quadratic function? (all ways)
For x0 :
1. x0 = −
b
- mainly for standard form
2a
y = a ( x − x0 ) + y0
2
2. In vertex form - ( x − x0 ) =
0
x = x0
y=
a ( x − x1 ) ( x − x2 )
x − x1 =
0
x = x1
3. In x-intercept form - x − x =
0
2
x = x2
x0 =
4. If f ( a) = f ( b ) , then x0 =
x1 + x2
2
a+b
2
For y0 mainly universal way is used:
1) find x0
2) plug x0 into the function instead of x and the result will be y0
39. How to find x-intercept of a function?
Substitute 0 instead of y. (Make y=0) (Find f ( x ) = 0 )
40. How to find y-intercept of a function?
Substitute 0 instead of x. (Make x =0) (Find f ( 0 ) )
41. Transformation of graphs to the left, right, up or down
If we move the function left or right, we change x in the function, i.e. the form of change is
always x-r.
If we move to the right, then this r is positive so we change x -> x-r. If we move to the left,
then this r is negative so we change x -> x+r.
If we move up or down, we change y in the function, i.e. the form of change is f(x) ± r.
If we move up, then we do f(x) +r. If we move down, then we do f(x) –r
Example: f(x)= 2x+3
Move 3 units to the right and 2 units up -> f(x-3) +2 => 2(x-3)+3+2 => 2x-1
Move 5 units to the left and 4 units down -> f(x+5) -4 -> 2(x+5) +3 -4 => 2x+9
42. Circle equation general form and how to find the center of the circle and radius?
R 2 - circle equation general form
( x − a) + ( y − b ) =
2
2
To find center equate each bracket to 0.
0
( x − a) =
x=a
0
( y − b) =
y=b
Center ( a;b )
Radius is provided in the right side of the equation. Put it under the square root and find it.
43. How to convert degrees into radians?
Multiply degrees by
π
180
44. How to convert radians into degrees?
Multiply radians by
180
or substitute 180 instead of π and simplify as far as possible.
π
45. What is sin, cos, tan, cot of an angle? (with drawing of a right triangle)
- SOHCAHTOA rule
Sin (Sine) is the ratio of opposite leg to hypotenuse
Cos (Cosine) is the ratio of adjacent leg to hypotenuse.
Tan (Tangent) is the ratio of opposite leg to adjacent leg.
*Cot (Cotangent) is the ratio of adjacent leg to opposite leg. It is not mentioned in
SOHCAHTOA rule.
46. What is the relationship between sin – cos and tan – cot in terms of 90° ?
If two angles add up to 90 degrees, then sine of one of these angles is equal to cosine of the
other.
Same rule applies to tangent and cotangent.
47. Signs of sin, cos, tan, cot in quadrants.
In I quadrant all trigonometric functions are positive
In II quadrant only sin (sine) is positive. The rest are negative.
In III quadrant only tan (tangent) and cot (cotangent) are positive. The rest are negative.
In IV quadrant only cos (cosine) is positive. The rest are negative.
48. Factor and remainder theorems of polynomials
If there is a root (solution, zero, x-intercept) r given, then the expression x − r is called the
factor of polynomial.
If we want to check whether x-r is a factor of a polynomial or not, we equate x-r = 0. Find x=r
and plug it in the polynomial for x. If the result is 0, it means that remainder is 0, and x-r is
factor. If the result of substitution is not 0, then the result obtained from substitution will be
remainder, and x-r will not be a factor of polynomial.
49. What is mean, mode, median and range? Ways of finding them.
Mean = sum of numbers divided by their amount
Mode = the most used number in set
Median = middle value in the data set.
If there are odd amount of data in the set, just divide the amount by 2 and round to the next
whole number = that will be the position of your median in the data set, written in increasing
or decreasing order. Then just find that value in the data set and it will be your median.
If there are even amount of data in the set, then divide the amount by 2 and take the
obtained and next value = these will be the positions between which you have your median.
Find the mean of these 2 numbers in your data set and you will have median.
Example:
Odd: 3,5,4,6,7  Move into order: 3,4,5,6,7  5 numbers in the set
5/2 = 2.5 ~~ 3rd  3rd number is 5. Median is 5.
Even: 3,5,4,6,7,8  Move into order: 3,4,5,6,7,8  6 numbers in the set
6/2 = 3  3rd and 4th  find the mean of them 
5+6
= 5.5 - median.
2
Range = the difference between the largest and the smallest value in a data set.
50. What is rate of change?
Slope of a line is also called the rate of change. It can be called average rate of change, or just
rate of change, and instead of “change” there can be increase or decrease.
51. What is standard deviation?
Standard deviation shows how far data are spread around the mean. The closer the data are
to each other, the less the standard deviation is going to be. (Desmos also available to check
stdev)
52. What is margin of error? Main way to reduce the margin of error.
Margin of error shows the maximum expected difference between the true data and random
sample. To reduce the margin of error the main way is to increase the sample size. The larger
the sample, the lower the margin of error
Example: Of 5000 randomly asked people out of 60000 75% think that sun is bright, with a
margin of error of 4%. This means that if we check for all 60000 people, the true value of data
will lie between 75%-4% and 75%+4% which means between 71% and 79%. So any value in
between these is a plausible value, and anything outside will not be reasonable/plausible.
53. Box – and – whisker diagram.
54. What is correlation? Explain it and give types of correlation.
Correlation = Connection. If the data are connected, then there is a correlation between them.
There are positive correlations, negative correlations and no correlations.
Positive correlations show an increasing function
Negative correlations show a decreasing function.
No correlation just shows the set of points which cannot be brought to a function.
55. What is the formula for probability?
p=
needed events
total events
pgeom =
needed area
total area
Geometry:
56. What is sum of interior and exterior angles of a polygon?
Interior:
Triangle = 180 degrees
Quadrilateral = 360 degrees
Pentagon = 540 degrees
Hexagon = 720 degrees
Any polygon = 180(n-2) degrees where n is the amount of sides.
Exterior angle sum is always 360, no matter the amount of sides.
57. Properties of supplementary, complementary and vertical angles
Supplementary angles sum is 180 degrees
Complementary angles sum is 90 degrees
Vertical angles are equal and they are formed from the intersection of straight lines. They also
form 4 pairs of supplementary angles. Total sum of all angles there is 360 degrees.
58. Relationship between the angles formed by two parallel
lines and a secant line.
Alternate angles are equal
One-sided angles add up to 180 degrees.
Corresponding angles are equal.
59. Any side of a triangle is always between the difference and sum of two other sides of this
triangle.
60. Largest side is opposite to the largest angle and smallest side is opposite to the smallest angle
in triangles.
61. Formulas for a regular (equilateral) triangle (perimeter, area, height, radius of inscribed and
circumscribed circle)
P = 3a
h=
a 3
2
A=
a2 3
4
=
R
=
r
a
3
a
2 3
R − radius of circumscribed circle
r − radius of inscribed circle
62. Pythagorean theorem
2
c=
a2 + b2 , where c is a hypotenuse, and a,b are legs of a right triangle.
63. Right triangle properties with 30° − 60° − 90° angles.
Leg opposite to 30 degrees is half the hypotenuse
Leg opposite to 60 degrees is 3 times the other leg
64. Right triangle properties with 45° − 45° − 90° angles.
Such triangles are called isosceles right triangles.
Legs are equal to each other.
Hypotenuse is 2 times the leg of a right triangle in this case.
65. How to find the area of a right triangle?
A=
1
ab
, where a,b are legs of a right triangle, OR A = chc , where c is a hypotenuse of a
2
2
triangle, and hc is a height drawn towards the hypotenuse.
66.
67. Similarity of triangles
Similar triangles have same angles, placed on the corresponding places. Sides are in a specific
ratio.
Ratio of sides, heights, medians, angle bisectors, perimeters, radii of inscribed and
circumscribed circles, circumferences of circles, etc. are all equal to some value of K.
Ratio of areas is equal to K2
Ratio of volumes is equal to K 3
If the ratio is 1:1, then the triangles are congruent (everything inside there is the same)
We can have only the similarity of the same class shapes (triangle-triangle, trapezoidtrapezoid, and so on)
68. Congruence of triangles.
There are 3 rules of congruence in triangles:
1. A-S-A (Angle-side-angle)
If there is a side and two adjacent angles to it in one triangle are correspondingly equal to a
side and two adjacent angles to it in another triangle, then these triangles are congruent
(equal)
2. S-A-S (Side-angle-side)
If there are two sides and angle in between them of one triangle are correspondingly equal to
two sides and an angle in between them of another triangle, then these triangles are
congruent (equal).
3. S-S-S (Side-side-side)
If three sides of one triangle are correspondingly equal to three sides of another triangle, then
these triangles are congruent (equal).
69. Inscribed and central angles of a circle.
Degree and radian measure of a central angle is equal to the degree and radian measure of a
corresponding arc to it in a circle.
Degree and radian measure of an inscribed angle is twice less than the degree and radian
measure of a corresponding arc to it in a circle.
70. Formulas for circle (Circumference and Area of circle)
C = 2πR
A = πR 2
71. Formula for finding the length of the arc of a circle.
=
larc
α
⋅ 2πR , α - central angle
360
If we have α in radians, we can use it, but will have to simplify 360 and π and will have the
following:
larc =
α
⋅ R = αR
1
72. Formula for finding the sector area of the circle.
α
⋅ πR 2 , α - central angle
360
A sec
=
tor
If we have α in radians, we can use it, but will have to simplify 360 and π and will have the
following:
α
⋅ R2
2
A sec tor=
73. How to find a segment of the circle?
A segment
= A sec tor − A triangle
Depending on what triangle we have there, we will use the corresponding formula for its area,
but it is always an isosceles one, as a minimum, because it is built on radii.
74. Formulas for parallelogram (perimeter, area)
=
P 2 ( a + b)
A = ah
A = ab sin α , where α - angle between the sides.
75. Formulas for rectangle (perimeter, area)
=
P 2 ( a + b)
A= a ⋅ b
2
d=
a2 + b2
76. Formulas for square (perimeter, area)
P = 4a
A = a2
d=a 2
77. Formulas for a regular hexagon (perimeter, area, height)
P = 6a
A= 6 ⋅
a2 3
, Like Area of equilateral triangle multiplied by 6.
4
78. Formula for a cylinder volume
V = πR 2h
79. Formula for a rectangular box (cuboid, rectangular prism) volume
V = lwh
80. Formula for a cone volume
V=
1
πR 2h
3
81. Formula for a cube volume
V = a3
82. Formula for a sphere volume
V=
4
πR 3
3
83. Formula for a pyramid volume
=
V
1
A
⋅ h , where Abase is the area of a shape that is the base of the pyramid. Can be
3 base
triangle, quadrilateral, or else.
84. Surface area formulas for rectangular box, cylinder, cube, sphere,
Rectangular box: A surface= 2 ( l ⋅ w + w ⋅ h + l ⋅ h)
Cylinder: A =
2πR 2 + 2πRh
surface
Cube: A surface = 6a2 , where a is the edge (side) of the cube. Note: Alateral = 4a2 - lateral surface
area of a cube
Sphere: A surface = 4πR 2
85. Density formula:
d=
m
, where m – mass of the object, v - volume
v
86. Density of population formula
dpopulation =
population
area