Quadratic Equations and Expressions - Mathematics Form 2 Notes

• Introduction
  • Expansion
  • The Quadratic Identities.
  • Factorization
• Quadratic Equations
  • Formation of Quadratic Equations
• Past KCSE Questions on the Topic.

Introduction

Expansion

• A quadratic is any expression of the form ax² + bx + c, a ≠ 0. When the expression (x + 5) (3x + 2) is written in the form, 3x² + 17x + 10,it is said to have been expanded

Example

Expand (m + 2n) (m−n)

Solution

Let (m − n) be a
Then (m + 2n)(m − n) = (m+2n)a
= ma + 2na
= m (m − n) + 2n(m−n)
= m^{2 −} mn + 2mn − 2n²
= m² + mn − 2n²

Example

Expand (¹/₄x − ¹/_x)²

Solution

(¹/₄x − ¹/_x)² = (¹/₄x − ¹/_x)(¹/₄x − 1/_x)
= ¹/₄(¹/₄x − ¹/_x) − ¹/_x(¹/₄x − 1/_x)
= ¹/₁₆ − ¹/_4x − ¹/_4x + ¹/_x²
= ¹/₁₆ − ¹/_2x + ¹/_x²

The Quadratic Identities.

 (a + b )² = (a² + 2ab + b²)

(a − b )² = (a^{2 −} 2ab + b²)
(a + b)(a − b) = (a² − b²)

Examples

(x + 2)² = x² + 4x + 4
(x − 3)² = x² − 6x + 9
(x + 2a)(x −2a) = x² − 4x²

Factorization

To factorize the expression , ax2 + bx + c ,we look for two numbers such that their product is ac and their sum is b. a , b are the coefficient of x while c is the constant

Example

8x² + 10x + 3

Solution

Look for two number such that their product is 8 x 3 = 24.
Their sum is 10 where 1 0 is the coefficient of x,
The number are 4 and 6,
Rewrite the term 10x as 4x + 6x, thus 8x² + 4x + 6x + 3
Use the grouping method to factorize the expression
= 4x (2x + 1) + 3(2x + 1 )
= (4x + 3) (2x + 1)

Example

Factorize
6x² − 13x + 6

Solution

Look for two number such that the product is 6 x 6 =36 and the sum is −13.
The numbers are −4 and –9
Therefore, 6x² − 13x + 6
= 6x² − 4x − 9x + 6
=2x(3x − 2) − 3(3x−2)
= (2x−3) (3x − 2)

Quadratic Equations

• In this section we are looking at solving quadratic equation using factor method.

Example

Solve x² + 3 x - 54 = 0

Solution

Factorize the left hand side
x² + 3x − 54 = x² − 6x + 9x − 54 = 0
Note;
The product of two numbers should be − 54 and the sum 3
=x² − 6x + 9x − 54
= x(x − 6)(x + 9) = 0
= (x − 6)(x + 9) = 0
x − 6 = 0, x +9 = 0
Hence x = −9 or x = 6

Example

Expand the following expression and then factorize it
(3x + y)² − (x − 3y)²

Solution

(3x + y)² − (x − 3y)² = 9x² + 6xy + y² −(x² − 6xy + 9y²)
=9x² + 6xy + y² − x² + 6xy + 9y²
= 8x² + 12xy − 8y²
= 4 (2x² + 3xy − 2y²)(You can factorize this expression further, find two numbers whose product is 4x²y² and sum is 3xy)
The numbers are 4xy and –ay
= 4(2x2 + 4xy - xy - 2y2)
= 4 [2x(x+2y) - y(x+2y)]
= 4 (x + 2y)(2x - y)

Formation of Quadratic Equations

Given the Roots

Given that the roots of quadratic equations are x = 2 and x = -3, find the quadratic equation
If x = 2, then x – 2 = 0
If x= −3, then x +3 =0
Therefore, (x – 2) (x + 3) =0
x² + x − 6 = 0

Example

A rectangular room is 4 m longer than it is wide. If its area is 12 m². find its dimensions.

Solution

Let the width be x m .its length is then (x + 4) m.
The area of the room is x (x+4)m²
Therefore x (x + 4) = 1 2
x² + 4 x = 12
x² + 4x − 12 = 0
(x + 6)(x − 2) = 0
(x+6) = 0 (x−2) = 0 therefore x = −6 or 2
−6 is being ignored because length cannot be negative
The length of the room is x + 4 = 2 + 4
= 6 m

Past KCSE Questions on the Topic.

1. Simplify (3mks)
   2y² − xy − x²
      2x²− 2y²
2. Solve the following quadratic equation giving your answer to 3 d.p. (3mks)
   ²³/_x − ¹/_x² − 120 = 0
   
3. Simplify (3 mks)
     16x²− 4     ÷  2x – 2
   4x²+ 2x − 2      x + 1
4. Simplify as simple as possible
   (4x+2y)² − (2y − 4x)²
   (2z + y)² − (y − 2x)²
5. The sum of two numbers x and y is 40. Write down an expression, in terms of x, for the sum of the squares of the two numbers.Hence determine the minimum value of x² + y²
6. Mary has 21 coins whose total value is Kshs 72. There are twice as many five shillings coins as there are ten shillings coins. The rest one shilling coins. Find the number of ten shilling coins that Mary has.
7. Four farmers took their goats to the market Mohamed had two more goats than Ali Koech had 3 times as many goats as Mohamed. Whereas Odupoy had 1 0 goats less than both Mohamed and Koech.
   1. Write a simplified algebraic expression with one variable. Representing the total number of goats
   2. Three butchers bought all the goats and shared them equally. If each butcher got 17 goats. How many did Odupoy sell to the butchers?