Lesson 1: Introduction to Quadratic Equations
Lesson 1: Introduction to Quadratic Equations
Learning Objectives
By the end of this lesson, you should be able to:
- Recognize and write quadratic equations in the standard form: ax2 + bx + c = 0.
- Factorise quadratic expressions using the "Product and Sum" method.
- Solve quadratic equations by factorisation.
Lesson Content
1. What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2. Its standard form is:
ax2 + bx + c = 0
Where a, b, and c are constants, and a ≠ 0.
Example: x2 - 5x + 6 = 0 (Here, a=1, b=-5, c=6)
2. The "Product and Sum" Method for Factorisation
To factorise a simple quadratic expression like x2 + bx + c, we need to find two numbers that:
- Multiply to give the Product (the last number, c)
- Add to give the Sum (the middle number's coefficient, b)
Let's look at a clear step-by-step example.
Example 1: Factorise and solve x2 + 7x + 12 = 0
Step 1: Identify the Product and the Sum
- Product = 12
- Sum = 7
Step 2: List the factor pairs of the Product (12)
Write down all the pairs of whole numbers that multiply to make 12:
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
Step 3: Add the pairs to find the Sum (7)
Now, add the numbers in each pair together to see which one equals our target sum of 7:
1 + 12 = 13 ❌
2 + 6 = 8 ❌
3 + 4 = 7 ✔️ (Correct!)
Step 4: Determine the correct signs
Before writing our brackets, we must check the signs based on our Product and Sum.
- Because the Product is positive (+12) AND the Sum is positive (+7), both of our factors must be positive.
(Teacher's Rule Reminder: If the product was positive but the sum was negative, both factors would be negative. If the product was negative, one factor would be positive and the other negative.)
Step 5: Write the factored form and solve
Using our correct numbers (+3 and +4), we write the equation in brackets:
(x + 3)(x + 4) = 0
According to the Null Factor Law, if two things multiply to give zero, at least one of them must be zero:
(x + 3)(x + 4) = 0
either x + 3 = 0 or x + 4 = 0
x = 0 - 3 or x = 0 - 4
x = -3 or x = -4
The solutions are x = -3 or x = -4.
3. Handling Negative Signs
Let's look at an example where the signs are different.
Example 2: Factorise and solve x2 - x - 12 = 0
Step 1: Identify the Product and the Sum
- Product = -12
- Sum = -1 (Remember, x is the same as 1x, so the coefficient is -1)
Step 2: List the factor pairs of 12
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
Step 3: Find the pair that gives a difference of 1 (since product is negative)
12 - 1 = 11 ❌
6 - 2 = 4 ❌
4 - 3 = 1 ✔️ (Correct numbers: 4 and 3)
Step 4: Determine the correct signs
- Because the Product is negative (-12), one factor must be positive and the other negative.
- Because the Sum is negative (-1), the larger number must be negative.
- Therefore, our factors are -4 and +3.
Step 5: Write the factored form and solve
(x - 4)(x + 3) = 0
(x - 4)(x + 3) = 0
either x - 4 = 0 or x + 3 = 0
x = 0 + 4 or x = 0 - 3
x = 4 or x = -3
The solutions are x = 4 or x = -3.
Example 3: Quick Working Flow
Solve x2 - 2x - 15 = 0
Product = -15, Sum = -2
Factors of 15: 1 × 15, 3 × 5
Difference: 5 - 3 = 2
Signs: (-5) + (+3) = -2
(x - 5)(x + 3) = 0
either x - 5 = 0 or x + 3 = 0
x = 0 + 5 or x = 0 - 3
x = 5 or x = -3
Solutions: x = 5 or x = -3
⚠️ Common MSCE Mistakes to Avoid
Forgetting the "= 0": You cannot factorise and solve if the equation is not set to zero.
→ Example: If given x2 + 5x = 6, you must first rewrite it as x2 + 5x - 6 = 0.
Sign Errors: Always double-check your signs in Step 4. Write out the addition/subtraction explicitly to avoid silly mistakes.
→ Example: If Product is negative and Sum is negative, the larger factor MUST be negative.
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Lesson 1 Practice: Introduction to Quadratic Equations
This formative assessment is designed to test your understanding of factorising and solving quadratic equations using the "Product and Sum" method, as outlined in the Form 3 Mathematics syllabus.
Instructions:
- For Multiple Choice questions, select the single best answer.
- For written questions, you must show your vertical stepped working clearly to get full marks.
- Total Marks: 15
- For Multiple Choice questions, select the single best answer.
- For written questions, you must show your vertical stepped working clearly to get full marks.
- Total Marks: 15
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