Mathematics

Quadratic Equation - Factorization of Quadratic Equation

len Alfred Ajibola - 11th February, 2019 @ 11:38 AM

Topics in Mathematics

Angles: Type of angles based on degree measurement Angle Pairs: Types of angle pairs Conversion of Number Bases explained with examples Angles: What are Angles? Categories of Angles Lines and Types in Mathematics Number Base System - Table showing the Number Bases with explanations Mathematics Scheme of Work, SS1, 3rd Term Mathematics Scheme of Work, SS1, 2nd Term Mathematics Scheme of Work, SS1, 1st Term Trigonometry - Sine, Cosine and Tangent explained with worked examples Quadratic Equation - Factorization of Quadratic Equation Algebra - Explanation and Worked Examples on Algebra


Academic Questions in Mathematics

Please check out our Test Your Knowledge page to see all Questions and Answers

Factorize 12a2 - 20a + 3

  • A. (6a - 1)(2a - 3)

  • B. (6a - 1)(2a + 3)

  • C. (6a + 1)(2a - 3)

  • D. (6a + 1)(2a + 3)

  • E. (-6a + 1)(2a - 3)

  • F. (-6a + 1)(2a + 3)

Simplify 10ab + 7ab

  • A. 17ab2

  • B. 17a2b2

  • C. 3ab2

  • D. 3a2b2

  • E. 17ab

  • F 3ab

Which of the following is not a type of angle?

  • A. Right angle

  • B. Left Angle

  • C. Acute Angle

  • D. Complete Angle

  • E. Straight Angle

  • F. Reflex angle

Two lines that can never meet each other are best considered as _____ lines.

  • A. straight
  • B. vertical
  • C. horizontal
  • D. parallel
  • E. oblique
  • F. perpendicular

An angle at exactly 90o is a/an _____ angle.

  • A. acute
  • B. obtuse
  • C. reflex
  • D. right
  • E. complete
  • F. straight

Slanted or tilted lines that will finally meet at some point but NOT at a right angle are referred to as _____.

  • A. Oblique
  • B. Curved
  • C. Vertical
  • D. Perpendicular
  • E. Parallel
  • F. Horizontal

The two sides that forms an angle are referred to as the _____.

  • A. Arms
  • B. Vertexes
  • C. Bays
  • D. Zones
  • E. Limbs
  • F. Dune

The point where the two sides of an angle meet is termed the _____.

  • A. Arms
  • B. Vertex
  • C. Bays
  • D. Zones
  • E. Limbs
  • F. Dune

LEN ACADEMY SMART SCHOOL SOFTWARE

Image

Read more on its smart academic features here

Please click here to kindly support education


Understanding Factorization:

Let’s start by explaining the reverse of factorization. Consider the equation below:

(4a + 6)(a – 1)

We can see 4 values from the above equation, they are:

4a, 6, a and -1

If we were to expand the brackets, we will have to:

  • Multiply +4a by a ….. '+4a x a' = 4a2

  • Multiply +4a by -1 ….. '+4a x -1' = -4a

  • Multiply +6a by a ….. '+6 x a' = +6a

  • Multiply +6a by – 1 ….. '+6 x -1' = -6


Note the followings:

  1. Minus x Plus = Minus
  2. Plus x Minus = Minus
  3. Minus x Minus = Plus
  4. Plus x Plus = Plus

Understand that when there isn't a sign in front of a number or variable, then it is considered to have a plus sign (+) before it.

If there isn’t any number in front of a variable, then it’s considered to have the value '1' before it. For instance ('a' has the same value as '1a').

The result from expanding (4a + 6)(a – 1) will give us:

(+4a x a) (+4a x -1) (+6 x a) (+6 x -1)

4a2 – 4a + 6a - 6

Since -4a and +6a are like terms (–4a + 6a will give us +2a), we will have our answer to be:

4a2 + 2a - 6


If you understood the above example, then factorization will become easier for you.

Factorization is simply the reverse of expanding brackets.

You may have to make some trials and errors when solving certain factorization questions, but following the steps in my examples will make factorization questions easier for you.

Please read on Algebra here

 

Example 1:

Factorize 4b2 – 10b + 4

We have three values from the question. They are:

+ 4b2, -10b, and +4

To factorize the above equation, we will need to split the middle value into 2 numbers whose:

  • Addition or Subtraction will be equal to -10b

  • Product will be equal to the 16b2 (that is, the product of 4b2 and 4 (at the right side).

(-8b – 2b) will work for us because:

  • Their subtraction = -10b

  • Their multiplication = 16b2

4b2 – 8b – 2b + 4

Notice that 4b2 and 8b are divisible by '4b', (that is, 4b is a factor) while 2b and 4 are divisible by 2, (that is, 2 is a factor). Therefore we can write:

4b(b - 2) -2(b - 2)

Now the brackets are the same, so our answer is:

Answer = (4b - 2)(b - 2)

 

Example 2:

Factorize 12a² - 20a + 3

To factorize the above equation, we will need to split the middle value into 2 numbers whose:

  • Addition or Subtraction will be equal to -20a

  • Product will be equal to 36a² (that is, the product of 12a2 and 3 (at the right side).

-18a – 2a will work for us because:

  • Their subtraction = -20a

  • Their multiplication = 36a²

12a2 – 18a – 2a + 3

Notice that 12a2 and 18a are divisible by '6a', (that is, 6a is a factor) while 2 and 3 are not divisible. Therefore we can write:

6a(2a - 3) -1(2a - 3)

Now the brackets are the same, so our answer is:

Answer = (6a - 1)(2a - 3)

Please read on the Number Base System here

 

Example 3:

Factorize d² + 2d - 8

To factorize the above equation, we will need to split the middle value into 2 numbers whose:

  • Addition or Subtraction will be equal to +2d

  • Product will be equal to the 8d² (that is, the product of d2 and 8 (at the right side).

+4d - 2d will work for us because:

  • Their addition = +2d

  • Their multiplication = 8d²

d2 + 4d - 2d + 8

Notice that d2 and 4d are divisible by 'd', (that is, d is a factor) while 2d and 8 are divisible by '2', (that is, 2 is a factor), Therefore we can write:

d(d + 4) -2(d + 4)

Now the brackets are the same, so our answer is:

Answer = (d - 2)(d + 4)

Please click here to follow Len Academy on Google News.

Please like and follow our official facebook page here for great educational write-ups.

You can follow Len Academy on twitter here.Thank you.


Kindly share this article via the links below:


len

Alfred Ajibola is a Medical Biochemist, a passionate Academician with over 7 years of experience, a Versatile Writer, a Web Developer, a Cisco Certified Network Associate and a Cisco CyberOps Associate.


Please Register here or Login here to contribute to this topic by commenting in the box below.

CONTRIBUTE TO THIS TOPIC | ASK A QUESTION


Amazing facts in Mathematics

The opposite sides of a 🎲 dice 🎲 will always give a sum of seven

31,556,926 seconds makes one year

From 0 through 1000, the only number that has the letter 'a' in it is 'one thousand'

÷

The division symbol '÷' is actually called an obelus

Four is the only number in the English language that is spelt with the same number of letters as the number (4) itself

40 is spelt as FORTY and not FOURTY. It is the only number that is spelt with letters arranged in alphabetical order 🙄

An odd number is a number that cannot be divisible by 2.

Every odd number has an 'e' in its spelling. They include:

  • One
  • Three
  • Five
  • Seven
  • Nine

555 in Thailand should sound like 'hahaha'. This is because the word '5' means 'ha' in Thailand

10! Actually means 10 factorial in mathematics. Below is its calculation:

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3628800

3628800 seconds is the same as 42 days

42 days is the same as 6 weeks

We can therefore say that 10! equals 6 weeks or 3628800 seconds in mathematics

When you multiply 111,111,111 by 111,111,111; the result equals 12,345,678,987,654,321


NOTABLE POINTS IN Mathematics

Sides of a right angle triangle - Len Academy

 

The image above shows a right angled triangle. The angle of the above right triangle is denoted as θ

 

Hypotenuse is always the longest side of the triangle.

Opposite is the side opposite the angle (θ)

Adjacent is the shorter side next to the angle (θ)

Each of Sin θ, cos θ and tan θ are the ratio of sides of the right triangle. They are calculated by using the formula (margic word) sohcahtoa:

Sohcahtoa - Len Academy

soh

Sin θ = Opposite/Hypotenuse

cah

Cos θ = Adjacent/Hypotenuse

toa

Tan θ = Opposite/Adjacent

Please read more on sine, cosine and tangent here

Circumference in mathematics means the distance around the edge of a circle (or a another curverd surface).

Perimeter means the distance around a square or rectangle or polygon.

Circumference is to circle or any other curved surface

Circumference of a Circle - Len Academy

Perimeter is to square or rectangle or polygon

Perimeter of a Rectangle - Len Academy