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ALGEBRA

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ALGEBRA

Algebra is one of the major branches of mathematics that deals with the use of symbols, letters, and numbers to represent quantities and relationships. It provides methods for solving problems in which some values are unknown.

By representing these unknown values with variables, algebra allows us to develop general rules and formulas that can be applied to many different situations.

In algebra, letters such as 𝑥, 𝑦 and 𝑧 are used as variables, which may stand for unknown numbers or quantities that can change. These variables are combined with numbers using mathematical operations to form algebraic expressions, equations, and inequalities.

CONTENTS:

Under this topic you will cover five broad concepts about Algebra as follows:

(i) Binary operations: Which involves combining two numbers or expressions using operations such as addition, subtraction, multiplication, or division.

(ii) Transposition of formulae: The process of rearranging equations to make a particular variable the subject.

(iii) Quadratic expressions: These are algebraic expressions of degree two usually written in the form 𝑎𝑥²+𝑏𝑥+𝑐.

(iv) Factorization of quadratic expressions: It involves writing such expressions as the product of two simpler expressions.

(v) Quadratic equations:This involves finding the values of the variable that satisfy equations of the form 𝑎𝑥²+𝑏𝑥+𝑐=0.

COMPETENCIES:

Upon completion of this chapter, you should demonstrate competencies in Algebra through:

(a) Understanding binary operations: Identifying and performing basic binary operations such as addition, subtraction, multiplication, and division on numbers and algebraic expressions.

(b)Transposing formulae: Rearranging mathematical formulae to make a required variable the subject of the formula.

(d) Factorizing quadratic expressions: Writing quadratic expressions into simpler algebraic factors.

(e) Solving quadratic equations: Finding the solutions to quadratic equations using appropriate methods such as factorization, completing the square, or applying the quadratic general formula.

(f) Applying algebraic skills in problem solving: Using algebraic methods to solve mathematical and real-life problems involving quadratic relationships.

These competencies will help you develop logical thinking and strong problem-solving skills that can be applied to many real-life situations.

They enable you to solve practical problems, such as those related to budgeting and personal finance, shopping and discounts, travel and time calculations, construction and measurement, business and profit analysis, as well as planning and scheduling.

In addition, mastering these competencies builds a solid foundation for higher mathematics and enhances your ability to think critically and apply mathematical ideas in everyday life.

Binary operations

A binary operation is a rule that combines two elements of a set to produce another element of the same set. It takes two inputs and gives one output, and the output must belong to the same set.

The basic tenets of binary operations and solve the related problems.

Explore the basic tenets of binary operations and solve the related problems.

In binary operations we use symbols like (∗ or 𝚫) called operators which may have different meanings in different contexts.

Therefore, these symbols (operators) must be defined either directly or indirectly. Their definition may include the basic mathematical operations such as addition (+), subtraction (−), multiplication (×) or division (÷).

Example 56

Given that 𝑎∗𝑏=3𝑎−𝑏, find the value of 5∗3.

Solution:

Example 57

If 𝐴∗B=A−2B, find:

Solution:

Example 58

Given that 𝑝∗𝑞=𝑝+𝑞+𝑝𝑞, find:

Solution:

Example 59

If 𝑀Δ𝑁=𝑀2−𝑁2, find 6Δ4.

Solution:

Example 60

Given that 𝑥∗𝑦=𝑥+2𝑦, find the value of:

Solution:

(b) 3∗(4∗2)

Example 61

Solution:

Exercise 13

1. Given that 𝐴∗𝐵=3𝐴+2𝐵, find:

2. If 𝑎Δ𝑏=2𝑎−𝑏, evaluate 5Δ3.

4. If 𝑎⋆𝑏=𝑎²−𝑏²+2𝑎𝑏, what is the value of 3⋆5?

7. It is defined that 𝑎⋆𝑏=𝑎²−𝑎𝑏+𝑏². Find 𝑥⋆(𝑥+1).

8. Given that 𝑎Δ𝑏=𝑎+𝑏² and 𝑎#𝑏=𝑎Δ𝑏+1, find (2Δ3)#4.

9. Let 𝑥∗𝑦=3𝑥−𝑦2, find the value of m if 5∗𝑚=6.

10. Given that 𝑠∗𝑡=2𝑠−𝑡, find (𝑠∗𝑡)∗(𝑠∗𝑡)

11. Let 𝑝∗𝑞=2−𝑝𝑞, what is the value of [(3∗1)∗(1∗3)]∗5

Transposition of formulae

Transposition of formulae is the process of changing the subject of a formula. It involves changing the position of variables in an equation so that the required variable stands alone on one side of the given equation.

For example, in the equation 𝑦=𝑥−3, y is the subject of the formula and it is standing alone in the left side of the equation.

From the equation 𝑦=𝑥−3, we can make 𝑥 the subject by adding 3 on each side of the equation, that is 𝑦+3=𝑥−3+3, which gives 𝑦+3=𝑥 or 𝑥=𝑦+3.

The concept of Transposition of formulae and solve the related problems

Describe the concept of Transposition of formulae and solve the related problems

When transposing a variable in a given formula (equation), you must apply inverse operations to isolate the required variable, therefore:

• Addition becomes subtraction

• Subtraction becomes addition

• Multiplication becomes division

• Division becomes multiplication

Also it is important to keep in mind that whatever you do to one side, do the same to the other.

Example 62

Given that P=2l+2w, make 𝑙 the subject.

Solution:

Example 63

Given that 𝑦=𝑚𝑥+𝑐, make m the subject of the formula.

Solution:

Example 64

Solution:

Exercise 14

In each of the following, make the given letter the subject of the formula:

Quadratic Expressions

A quadratic expression is an algebraic expression in which the highest power of the variable is 2.These expressions take the form of 𝑎𝑥²+𝑏𝑥+𝑐, where:

• 𝑎, 𝑏, and 𝑐 are constants (real numbers)

• 𝑥 is the variable

• 𝑎≠0 (if 𝑎=0, it is no longer quadratic)

Examples of quadratic expressions include 𝑥²+5𝑥+6, 3𝑥²−7𝑥+1, 4𝑦²−9, 𝑥²+2𝑥, 𝑥²−8 and 3𝑥².

The basic tenets of quadratic expressions.

Explore the basic tenets of quadratic expressions.

How do quadratic expressions arise in mathematics?

Quadratic expressions arise naturally whenever a situation involves a square of a variable, that is, when a quantity is multiplied by itself.

For example, in measuring the area (A) of a rectangular playing pitch, we multiply the Length(L) by Width (W), that is, 𝐴=𝐿×𝑊.

If the pitch has a length of 2𝑥 and width 𝑥 units as shown in figure 4.1 below, then its area is given by 𝐴=𝐿×𝑊=2𝑥×𝑥=2𝑥².

Suppose that the length of the pitch is increased by 5 and the width by 4 units, then its new dimensions are (2𝑥+5) and (𝑥+4) as shown in figure 4.2 below:

From figure 4.2 above, the new area of the pitch can be found as follows:

Therefore, 2𝑥² and 2𝑥²+13𝑥+20 are quadratic expressions representing areas of rectangles having (2𝑥 length and 𝑥 width) and (2𝑥+5 length and 𝑥+4 width) respectively.

Quadratic expressions also arise when you multiply any two linear expressions, for example(𝑥+2)(𝑥+5), which upon expansion gives 𝑥²+7𝑥+10.

Example 65

Write (𝑥−2)² in expanded form.

Solution:

Example 66

Solution:

Example 67

Write the middle term from the [removed]2𝑦+6)(3𝑦−5).

Solution:

Example 68

A rectangular garden has length (𝑥+5)meters and width (𝑥−3)meters, write and simplify the expression for its area.

Solution:

Basic Quadratic Expressions Identities

The following are products which automatically create quadratic expressions

NB: From the general quadratic [removed]ax²+bx+c), if:

Exercise 15

1. Write each of the following expressions in expanded form:

4. Simplify each of the following expressions:

5. Write the middle terms of each of the following expressions:

6. Use the identity (𝑥+𝑎)(𝑥−𝑎)=𝑥²−𝑎² to find the exact value of each of the following numbers:

7. The length of a rectangle is 𝑥 and the width is (12−𝑥),write and simplify the expression for its area.

8. A company sells an item at 𝑥 dollars each. If sells (100−2𝑥) items per day, write an expression for the daily revenue and simplify it.

9. Two consecutive numbers are such that the second number exceeds the first by 5, write an expression for their product in a simplified form.

Factorization of Quadratic Expressions

Factorization of a quadratic expression simply means rewriting it as a product of two linear expressions. It involves finding the factors of the given quadratic expression.

The concept of factorization of quadratic expressions

Explain the concept of factorization of quadratic expressions.

In algebra, expressions are treated like common numbers, for example, the number 6 can be written as 1×6 or 2×3, therefore, 1, 2, 3 and 6 are the factors of 6.

Similarly, 𝑥²−3𝑥−10=(𝑥+2)(𝑥−5) , so (𝑥+2) and (𝑥−5) are the factors of𝑥²−3𝑥−10.

Factorization Methods

Quadratic expressions can be factorized by using the following common methods:

(i) Splitting the Middle Term (ii) Difference of Two squares (iii) Perfect squares

Factorization by Splitting the Middle Term

This method involves writing the middle term ′𝑏𝑥′ into two terms so that we can find the factors by grouping.

Steps:

(i) Identify 𝑎 and 𝑐 then find their product

(ii) Find the possible two numbers (factors of 𝑎𝑐) that multiply to 𝑎𝑐 and add to 𝑏.

Example 69

Factorize 𝑥²+9𝑥+20 by splitting the middle term:

Solution:

Example 70

Factorize 2𝑥²+7𝑥+3 by splitting the middle term.

Solution:

Example 71

Factorize 2𝑥²+11𝑥+5 by splitting the middle term.

Solution:

Example 72

Factorize 𝑥²−2𝑥−15 by splitting the middle term.

Solution:

Exercise 16

Factorize each of the following quadratic expressions by splitting the middle term:

14. The area of a rectangular garden is given by (3x²+10x+7) Sq. meters, what are the possible dimensions of the garden?

15. The product of two consecutive numbers is given by (y²−7y+12), what are the numbers?

The different methods of factorizing quadratic expressions.

Explore the different methods of factorizing quadratic expressions.

Factorization of Difference of Two squares

The method is used to factorize quadratic expressions in the form 𝑥²−𝑎² (the difference of two squares).

Remember that (𝑥+𝑎)(𝑥−𝑎)=𝑥²−𝑎𝑥+𝑎𝑥−𝑎²=𝑥²−𝑎²

So x²−a²=(x+a)(x−a) means (𝑥+𝑎) and (𝑥−𝑎) are the factors of 𝑥²−𝑎².

Example 73

Factorize 𝑥²−36.

Solution:

Example 74

Factorize 9𝑥²−25

Solution:

Example 75

Find the exact value of 1001²−999².

Solution:

Example 76

Find the exact value of 500²−497².

Solution:

Exercise 17

1. Factorize each of the following expressions:

2. Factorize each of the following expressions:

3. Simplify each of the following numbers:

Factorization of perfect squares

Perfect squares are quadratic expressions formed by squaring a single linear expression, therefore the have two identical factors.

They are written in the form of (𝑥+𝑎)² or (𝑥−𝑎)².

When factorizing perfect squares, the middle term always splits into two equal terms which add up to 2𝑎 or −2𝑎.

Example 77

Factorize 𝑥²+6𝑥+9.

Solution:

Example 78

Factorize 4𝑥²−12𝑥+9

Solution:

Example 79

Factorize 𝑦²−𝑦+1/4.

Solution:

Exercise 18

1. Factorize each of the following perfect squares:

2. Find the value of 𝑘 that makes each of the following expressions perfect squares:

3. Which of the following expressions are perfect squares?

4. A farmer owns a square piece of land whose area is (𝑝²−24𝑝+9) square meters, what is the side length of the land?

5. The area of a square storage room is (36𝑝²+12𝑝+1) square meters, find the length of its one side.

Quadratic Equations

A quadratic equation is an equation in which the highest power of the variable is 2, so it has the general form 𝑎𝑥²+𝑏𝑥+𝑐=0, where:

Examples of such equations include 𝑥²+5𝑥+6=0, 2𝑥²−7𝑥+3=0, 4𝑥²=9 and (𝑥−3)²=16.

Solve quadratic equations by using different methods (factorization, Completing the square and quadratic formula).

Solve quadratic equations by using different methods (factorization, Completing the square, and quadratic formula).

Solving Quadratic Equations by Factorization Method

Solving quadratic equations by factorization method involves the following steps:

(1) Write the equation in standard form:

Make sure the equation is written as 𝑎𝑥²+𝑏𝑥+𝑐=0 (all terms must be on one side).

(2) Factorize the quadratic expression:

Write the expression 𝑎𝑥²+𝑏𝑥+𝑐 as a product of its two factors.

(3) Let each factor equals to zero and solve for the unknown.

Note that, if (𝑥+𝑘₁)(𝑥+𝑘₂)=0, then either (𝑥+𝑘₁)=0, or (𝑥+𝑘₂)=0 or both factors equal to 0, this is called the Zero Product Rule.

Example 80

Solve for x in the equation (𝑥+2)(𝑥−3)=0.

Solution:

Example 81

Solve the equation 𝑥(𝑥−5)=0

Solution:

Example 82

Use the factorization method to solve the equation 𝑥²−9𝑥+18=0.

Solution:

Example 83

Solve the equation 4𝑥²+12𝑥+9=0 by factorization method.

Solution:

Example 84

Solve the equation 𝑥²−2𝑥−15=0 by factorization method.

Solution:

Example 85

Solve the equation 15y²−y=2 by factorization method.

Solution:

Example 86

Solve the equation 𝑥²=81 by factorization method.

Solution:

Example 87

The length of a rectangle is 3m more than its width, if the area of the rectangle is 40m², what are the dimensions of the rectangle?

Solution:

Now 𝑥²+3𝑥−40=(𝑥+8)(𝑥−5)=0, which implies that either 𝑥+8=0 or 𝑥−5=0;

Exercise 19

1. Solve the following quadratic equations by using factorization method:

2. The product of two consecutive integers is 56, find the two integers.

3. The base of a triangle is 4 cm longer than its height, if the area of the triangle is 48 cm², find the base and height.

4. The product of two consecutive even integers is 168, find the integers.

5. The side of a square is reduced by 3 meters, if the new area is 40 m² less than the original area, find the original side length.

6. If the sum of two numbers is 15, and their product is 56, what are the two numbers?

7. The perimeter of a rectangle is 34 m, and its area is 60 m². Find its dimensions.

Solving Quadratic Equations by Completing the Square Method

Completing the square is another method used to solve quadratic equations that involves transformation of the left-hand side of the equation into a perfect square.

Procedures: