Algebraic Expressions and Equations

🎯 Lesson Objectives

By the end of this lesson, students should be able to:

• Simplify algebraic expressions
• Expand and factorize expressions
• Solve linear and quadratic equations

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📖 1. Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations.

Examples:

• $4x+7$
• 3×2−5x+23x^2 – 5x + 23x2−5x+2

🔹 Simplifying Expressions

Combine like terms:

$$3x+2x=5x$$

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📖 2. Expanding Brackets

Use multiplication to remove brackets.

Example:

(x+3)(x+2)=x2+5x+6(x + 3)(x + 2) = x^2 + 5x + 6(x+3)(x+2)=x2+5x+6

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📖 3. Factorization

Factorization is the reverse of expansion.

Example:

x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3)

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📖 4. Solving Linear Equations

A linear equation has the highest power of 1.

Example:

$$2x+6=14$$

Solve:

$$2x=8\Rightarrow x=4$$

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📖 5. Solving Quadratic Equations

A quadratic equation has the form:

ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0

🔹 Method 1: Factorization

x2+5x+6=0⇒(x+2)(x+3)=0x^2 + 5x + 6 = 0 \Rightarrow (x + 2)(x + 3) = 0x2+5x+6=0⇒(x+2)(x+3)=0 $$x=-2\text{or}x=-3$$

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🔹 Method 2: Quadratic Formula

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🌍 Real-Life Application

• Calculating profit and loss
• Determining distances and speeds
• Solving everyday numerical problems

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📝 Practice Exercises

1. Simplify: $5x+3x-2$
2. Expand: $(x+4)(x-1)$
3. Factorize: x2+7x+10x^2 + 7x + 10x2+7x+10
4. Solve: $3x+5=20$
5. Solve: x2+6x+8=0x^2 + 6x + 8 = 0x2+6x+8=0

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💡 Lesson Summary

Students learned how to manipulate algebraic expressions and solve equations, which are essential skills for higher-level mathematics.

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✅ Lesson 2: Introduction to Functions and Graphs

🎯 Lesson Objectives

By the end of this lesson, students should be able to:

• Understand what a function is
• Evaluate simple functions
• Identify basic types of functions
• Draw and interpret simple graphs

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📖 1. What is a Function?

A function is a rule that assigns one output to each input.

Example:

$$f(x)=2x+3$$

If $x=2$:

$$f(2)=7$$

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📖 2. Linear Functions

Linear functions produce straight-line graphs.

• $m$ = slope (steepness)
• $b$ = y-intercept

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📖 3. Quadratic Functions

Quadratic functions form curved graphs (parabolas).

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📖 4. Drawing Graphs (Basic Steps)

1. Choose values of $x$
2. Calculate $y$
3. Plot points
4. Join points smoothly

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📖 5. Domain and Range (Basic Idea)

• Domain → values of $x$
• Range → values of $y$

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🌍 Real-Life Application

• Tracking business growth
• Predicting population increase
• Understanding motion in physics

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📝 Practice Exercises

1. Find $f(4)$ if $f(x)=3x+1$
2. State whether this is linear or quadratic: y=x2+2y = x^2 + 2y=x2+2
3. Draw a table for $y=x+2$ when $x=0,1,2$
4. Identify slope and intercept in $y=2x+5$

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💡 Lesson Summary

Students learned how functions describe relationships between variables and how graphs help visualize