Specific Objectives
By the end of the topic, the learner should be able to:
- Expand algebraic expressions that form quadratic equations.
- Derive the three quadratic identities.
- Identify and use the three quadratic identities.
- Factorize quadratic expressions including the identities.
- Solve quadratic equations by factorization.
- Form and solve quadratic equations.
Content
- Expansion of algebraic expressions to form quadratic expressions of the form
ax2 + bx + c, where a, b, and c are constants. - The three quadratic identities:
Introduction
Expansion
A quadratic expression is any expression of the form ax2 + bx + c, where a ≠ 0. When an expression such as (x + 5)(3x + 2) is written in this form, it is said to have been expanded. Expansion involves multiplying the terms in the brackets and simplifying to get a quadratic expression.
Example
Expand (m + 2n)(m – n)
Solution
Let (m – n) be denoted as a.
Then (m + 2n)(m – n) = (m + 2n)a
= ma + 2na
= m(m – n) + 2n(m – n)
= m2 – mn + 2mn – 2n2
= m2 + mn – 2n2
Example
Expand (x + 3)(x – 4)
Solution
= x(x – 4) + 3(x – 4)
= x2 – 4x + 3x – 12
= x2 – x – 12
The Quadratic Identities
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 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 – 4a2
Factorization
To factorize a quadratic expression, we look for two numbers such that their product is ac and their sum is b. Here, a and b are the coefficients of x2 and x respectively, while c is the constant term.
Example
Solution
Look for two numbers whose product is 8 × 3 = 24.
Their sum is 10, which is the coefficient of x.
The numbers are 4 and 6.
Rewrite the term 10x as 4x + 6x, thus:
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 numbers such that their product is 6 × 6 = 36 and their sum is -13.
The numbers are -4 and -9.
Therefore,
= 2x(3x – 2) – 3(3x – 2)
= (2x – 3)(3x – 2)
Quadratic Equations
In this section, we focus on solving quadratic equations using the factor method. This involves expressing the quadratic equation in a factorized form and then solving for the variable.
Example
Solve x2 + 3x – 54 = 0
Solution
Factorize the left-hand side.
Note: The product of two numbers should be -54 and their sum 3.
The numbers are 9 and -6.
Therefore, (x – 6)(x + 9) = 0.
Hence, x – 6 = 0 or x + 9 = 0.
So, x = 6 or x = -9.
Example
Expand the following expression and then factorize it:
Solution
= (2xy – ay)(2xy + ay)
= (2xy)2 – (ay)2
= 4x2y2 – a2y2
(You can factorize this expression further. Find two numbers whose product is the above expression.)
The numbers are 4xy and –ay.
Formation of Quadratic Equations
Given the Roots
Given that the roots of a quadratic equation 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.
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).
Therefore, x(x + 4) = 12.
x2 + 4x – 12 = 0.
Factorize:
(x + 6)(x – 2) = 0.
-6 is ignored because length cannot be negative.
The length of the room is x + 4 = 2 + 4 = 6 m.
End of topic
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Past KCSE Questions on the Topic
- Simplify
(3 marks) - Solve the following quadratic equation giving your answer to 3 d.p. (3 marks)
- Simplify (3 marks)
16x2 – 4 ÷ 2x – 2
4x2 + 2x – 2x + 1 - Simplify as simply as possible
- 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.
- 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 are one shilling coins. Find the number of ten shilling coins that Mary has.
- 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 10 goats less than both Mohamed and Koech.
- Write a simplified algebraic expression with one variable representing the total number of goats.
- Three butchers bought all the goats and shared them equally. If each butcher got 17 goats, how many did Odupoy sell to the butchers?
(3 marks)
