Based on your uploaded chapter on quadratic equations.

1. What is a Quadratic Equation?

A quadratic equation has the form:

 ax^2 + bx + c = 0

ax^2 + bx + c = 0

Where:

• (a, b, c) are numbers
• (a ≠ 0)

Examples:

x^2 + 5x - 6 = 0

2x^2 - 7x + 3 = 0

2. Why Quadratics Matter

Quadratics model:

• ball trajectories
• arches
• bridges
• jumping/diving paths
• maximum heights
• area problems

The chapter overview discusses real-world uses including sports, architecture and satellite dishes.

3. The Shape — The Parabola

Quadratics graph as a parabola.

Positive (a)

Parabola opens upward.

Negative (a)

Parabola opens downward.

4. Solving by Factorising

Core Idea

If:

(x - 3)(x + 2) = 0

then:

x - 3 = 0
\quad \text{or} \quad
x + 2 = 0

So:

x = 3
\quad \text{or} \quad
x = -2

The Null Factor Law

If two numbers multiply to make zero:

ab = 0

then:

a = 0
\quad \text{or} \quad
b = 0

This is the MOST IMPORTANT algebra rule in factorising quadratics.

5. Factorising Patterns

Simple Trinomial

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

Find:

• two numbers multiplying to (+6)
• adding to (+5)

Difference of Squares

x^2 - 25 = (x - 5)(x + 5)

Pattern:

a^2 - b^2 = (a - b)(a + b)

6. Solving Using the Quadratic Formula

When factorising is hard, use:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The textbook introduces this in Lesson 8.3.

Example

Solve:

x^2 - 5x + 6 = 0

Identify:

• (a = 1)
• (b = -5)
• (c = 6)

Substitute carefully.

7. The Discriminant

The discriminant is:

\Delta = b^2 - 4ac

It tells us HOW MANY solutions exist.

Discriminant Summary

From Lesson 8.5.

Visual Meaning

• Crosses x-axis twice → 2 solutions
• Touches once → 1 solution
• Misses x-axis → no real solutions

8. Solving Graphically

Solutions are where the graph crosses the x-axis.

Example:

y = x^2 + x - 2

Solutions:

x = 1
\quad \text{and} \quad
x = -2

The textbook explains that the x-intercepts are the solutions.

9. Completing the Square

Used when factorising is difficult.

Example:

x^2 + 6x + 2 = 0

Add and subtract:

\left(\frac{6}{2}\right)^2 = 9

So:

x^2 + 6x + 9 - 7 = 0

Factorise:

(x + 3)^2 - 7 = 0

Then solve.

The textbook explains this process in Lesson 8.2.

10. Worded Problems

Typical Year 10 applications:

• area
• projectile motion
• dimensions
• consecutive numbers
• geometry

Example Structure

1. Define variable
2. Build equation
3. Rearrange to zero
4. Solve
5. Check answer
6. Write sentence with units

Example

Two consecutive numbers multiply to make (20).

Let first number be:

x

Second number:

x + 1

Equation:

x(x + 1) = 20

Expand:

x^2 + x - 20 = 0

Factorise:

(x + 5)(x - 4) = 0

Solutions:

x = -5
\quad \text{or} \quad
x = 4

So the pairs are:

• ((-5, -4))
• ((4, 5))

Based on Worked Example 4.

11. Common Exam Mistakes

Forgetting:

• to rearrange equation to (=0)
• negative signs
• both solutions
• brackets
• square roots
• checking factor pairs

12. High Value Exam Questions

You should be able to:

✅ Factorise quadratics
✅ Use Null Factor Law
✅ Use quadratic formula
✅ Solve graphically
✅ Use discriminant
✅ Solve worded problems
✅ Interpret graphs

13. Memory Tricks

Quadratic Formula

“Minus b, plus or minus square root…”

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant

Positive → 2 solutions
Zero → 1 solution
Negative → none

14. Essential Summary

Quadratics are about:

• curved relationships
• solving for unknowns
• graph behaviour
• real-world motion and area

The four BIG skills are:

1. Factorising
2. Quadratic Formula
3. Graphs
4. Discriminant

Useful worked examples, exercises and graphs appear throughout your uploaded chapter.