Finding quadratic equations of given roots.

 Find the quadratic equations of the following:

1. -2 and 3

2. 1/2 and 1

3. +√5 and -√5

Solutions:

1. -2 and 3 

Let x= -2 and x= 3

X+2=0 and x-3=0

Put the two equations into two different brackets and equate all to 0

(X+2)(X-3)=0

Expand brackets(remove brackets by multiplying every term in the first bracket by each term in the second bracket)

X(X-3)+2(X-3)=0

x² -3x + 2x -6=0

x² - x - 6=0

2. ½ and 1

Let x=½ and x=1

x-½=0 and x-1=0

To eliminate the fraction in the first equation, multiply by 2

2x-1=0 and x-1=0

(2x-1)(x-1)=0

Expand

2x(x-1)-1(x-1)=0

2x² -2x-x+1=0

2x²-3x+1=0

3. +√5 and -√5

Let x=+√5 and x=-√5

x-√5=0 and x+√5=0

(x-√5)(x+√5)=0

Expand

x(x+√5)-√5(x+√5)=0

x²+x√5-x√5-5=0

x²-5=0

Please, feel free to ask your question if any.

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Expect more!

God bless you all.

Differentiations.

For the purpose of those who are preparing for WAEC and UTME, I will like to introduce you to the Mathematics called, "Derivatives"
Do not be scared as I am going to make it so simple for you.
In differentiation, the derivative of any constant is Zero, the derive of X is 1.
There are different rules for finding derivatives in Calculus.
To find the derivative of a number is like finding the square root of a number.
The Rules of derivatives are:
1. Power Rule
2. Quotient Rule
3.Product Rule
4.Chain Rule.
We shall go through them one after the other.
  The Power Rule
Find the derivative 
f(x)=x⁴+2x³-x²+4x-1
To find the derivative of the above function, find the derivative of each term individually. In doing so, make the power of x the coefficient of x and reduce the power by 1 while, in terms where x has a coefficient, multiply the power of x by its coefficient and reduce the power by 1.Also to remember is that any constant is Zero.
   Solution to our problem
  Formula is: 

f(x)=x⁴+2x³-x²+4x-1
f'(x)=4x³+6x²-2x+4
Stay tune for the  other rules.
Until I come your way again, stay safe!

Welcome back!

Now, we are going to look at Quotient Rule.

Quotient Rule is applied when one function is divided by another function

You can easily remember Quotient rule by: 

This is read as LowDiHigh minus HighDiLow all over LowLow.

It means the under function multiply by the derivative of the on top function minus the on top function multiply by the derivative of the under function all divided by the square of the under function.

For more clarification, look at this: 

Multiply 3x-1  by the derivative of x² minus x² multiply by the derivative of 3x-1 all over (3x-1)²

See below for more clarification: 

The derivative of x²=2x

The derivative of 3x-1=3

Since 1 is a constant, its derivative=0

Expect other rules later in the day.

Angles

Angles are formed in different ways.

Follow me to see the different ways Angles are formed.

1. Angle can be formed on a straight line. An angle formed on a straight line is 180°,
2. When a vertical line is drawn perpendicularly to a horizontal line, a right angle is formed. A Right Angle is simply angle 90°,
3. When a transversal line is drawn across two parallel lines,  corresponding and alternate angles are formed. Corresponding angles are equal. Alternate angles are also equal. Parallel lines are lines that can never meet even if you draw them from Earth to Heaven. Corresponding angles are otherwise known as F-Shape angles while alternate angles are Z-Shape angles.
4. When two transversal lines cut across themselves, vertically opposite angles are formed and vertically opposite angles are equal.
5. Complimentary angles are angles that add up to 90°, e.g 60° + 30°, 45° + 45° etc.
6. Supplementary angles are angles that add up to 180°, e.g 100°+80°,20°+160°,90°+90°, etc
7. Angles formed at a point, add up to 360°,
8. In a triangle, the total sum of angles is 180°.

Angles

What are Angles?

When two or more lines meet at a point, angles are form. Angles are a measure of the degree of turning. There are different forms of angles. 

Namely:

1. Acute Angles: These are angles that are less than 90°, e.g 10°, 20°, 45°, 89°
2. Right Angle: This is an angle of 90°
3. Obtuse Angle: An angle that is more than 90° but less than 180°, e.g 91°,95°,179°,
4. Reflex Angles: They are angles that are greater than 180° but less than 360°,e.g 182°, 185°, 270°, 359°.

Guys, I am feeling sleepy. Please, see you soon.

Names of different triangles.

This may seem very simple but it is necessary for you to know the types of triangles to avoid embarrassment.

An undergraduate in,"Who Wants To Be A Millionaire " could not tell the name of a triangle with all its sides unequal some years ago. Therefore, the importance of knowing the names of triangles.  Viz:

1. Equilateral Triangle: This is a triangle with all equal sides and all equal angles,

2. Scalene Triangle: This is a triangle with all unequal sides and unequal angles,

3.Isosceles Triangle: This is a triangle with equal opposite sides and equal base angles,

4.Right Angle Triangle: This is a triangle which has 90° as one of its angles.

Names of Polygons from 1-20

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• Meaning of polygon:

A polygon is any plane shape bounded by straight lines. 

Polygons are named according to their number of sides.

Namely:

3 Triangle: Is a three sided polygon,

4 Quadrilateral: Is a four sided polygon,

5 Pentagon:  Is a five sided polygon,

6 Hexagon: Is a six sided polygon,

7 Heptagon: Is a seven sided polygon,

8 Octagon: Is an eight sided polygon,

9 Nonagon is a nine sided polygon,

10. Decagon: Is a ten sided polygon,

11 Endecagon:Is an eleven sided polygon,

12 Dodecagons:Are twelve sided polygons,

13 Tridecagon: Is a thirteen sided polygon,

14 Tetradecagon: Is a fourteen sided polygon,

15 Pentadecagon: Is a fifteen sided polygon,

16 Hexadecagon: Is a sixteen sided polygon,

17 Heptadecagon : Is a seventeen sided polygon,

18 Octadecagon: Is an eighteen sided polygon,

19 Enneadecagon: Is a nineteen sided polygon, 

20 Icosagon : Is a twenty sided polygon.

How to find the total sum of angles in any polygon.

Whenever you are asked to calculate the total sum of angles in any polygon of any number sides, please use this formula:

(n-2)×two right angles. One Right Angle is 90°, therefore, two Right Angles is 180°.

In simple form, the formula for finding the total sum of angles of an n-sided polygon is:

(n-2)×180. Where  n=number of sides of any  given polygon.

For instance: Find the total sum of angles of a Pentagon.

Solution:

Remember, a Pentagon is a 5-sided polygon.

Therefore: (n-2)×180=(5-2)×180°

3×180°=540°

Ans=540°

You can calculate the total sum of angles of any polygon using that formula,(n-2)×180°

 Stay tune for more.

If you have any question, please drop it on comment section.

Logarithm

First let's understand what Logarithm of a number is. The log of a number is the exponent or power or index to which the base of that number must be raised  to give you the number in question.

Example:

log1000 to base 10=3

When a log of a number is written without a base, its base is 10. Remember, we said that, the log of a number is the exponent or power or index of its base which will give the number in question. In this case the number is 1000 to base 10. So, the power to which 10 must be raised to give 1000 is 3. Therefore, the log of 1000 to base 10 is 3.

Expect more on Logarithm.

Expressing exponential numbers in Logarithm:

Having understood the meaning of logarithm, it is easy to express numbers in Index form  in Logarithm.

Example:

Express 8²=64  in Logarithmic form.

                  Solution:

8²=64 

In the given number above, 8 is known as the base while 2 is the Index or exponent or power. The 2 is the Logarithm of 64 to base 8 and it is written as:

Standard form(Scientific notation)

Standard form is a number of the form, A×10^n

Where A is between 1 and 10, i.e, 1 2 3 4 5 6 7 8 9 while n is the number of places the decimal point is moved to the left or right.
When the decimal point is moved to the left, n is positive while it is negative when moved to the right.
When a whole number is written, it has its decimal point at the end.
E.g write in standard form 2567653
Solution:
2567653
To make this number fall between 1 and 10, the decimal point has to be between 2 and 5, therefore, 2567653=2.567653×10^6.
Note that the exponent is positive because the decimal point is moved to the left.
Express in standard form, 0.025
Solution:
When expressing a decimal fraction in standard form, always moved the decimal point to the right such that the whole number is between 1 and 10 and the power must be negative in correspondence to the number of places the decimal point is moved to the right.
So, 0.025=2.5×10^-2

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  Please carefully study these for easier understanding of my next lesson. See you soon!  When expressing numbers in standard form or scientific notation, always remember that, when  the decimal point is moved to the right, the power or index or exponent must be negative  while if it is moved to the left, the power or Index or exponent must be positive. See example below: