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📘 Algebra • Lesson 15 of 22

Quadratic Equations

A quadratic equation contains a variable whose highest power is 2 . In this lesson, you will learn how to solve quadratic equations using factorisation, completing the square and the quadratic formula.

✏️ Algebra 📘 Worked examples included 🧠 Step-by-step learning

Learning Objectives

You should be able to:

Introduction

This lesson develops Quadratic Equations from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.

Key Notes

Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.

🎬 Roots are where the graph meets the axis

At each root, the quadratic has value zero.

🤔 Think About This...

Imagine throwing a ball into the air.

The ball rises, reaches its highest point and then falls back down. Its path forms a curve called a parabola.

Quadratic equations can help us calculate important points on that path, such as when the ball reaches the ground.

Quadratic equations are used to model curved paths, areas, motion and many other real-life situations.

📖 What is a Quadratic Equation?

A quadratic equation is an equation whose highest power of the variable is 2.

The three equations below are quadratic:
x² + 5x + 6 = 0
2x² − 7x + 3 = 0
5x² = 20
The two equations below are not quadratic:
2x + 5 = 0
7 = 0
A quadratic equation is usually written in the standard form:

ax² + bx + c = 0

where a ≠ 0.

🧠 The StudyNest Thinking Method

Before solving, ask:

👉 Is the equation equal to zero, and which method will be easiest?
  1. Make sure one side of the equation is equal to zero.
  2. Look at the quadratic carefully.
  3. Decide whether it factorises easily.
  4. If it does not factorise easily, consider completing the square or using the quadratic formula.
  5. If a particular method is requested, use that method.

🎯 Choosing the Best Method

Different quadratic equations may be easier to solve using different methods.

If you notice... A useful method to try
The quadratic factorises neatly Factorisation
You need to rewrite the quadratic in square form Completing the square
The quadratic does not factorise easily Quadratic formula
The question tells you to use a graph Graphical method
🎯 Exam Tip:

Unless the question gives a specific method, use the method that gives the clearest and shortest correct working.

1️⃣ Method 1 — Solving by Factorisation

Factorisation works well when the quadratic can be written as two simple brackets.

Steps for solving by factorisation

1. Write the equation equal to zero.
2. Factorise the quadratic expression.
3. Set each bracket equal to zero.
4. Solve both simple equations.
5. Check both answers.

Example 1

Solve:

x² + 7x + 10 = 0

Step 1: The equation is already equal to zero.

Step 2: Factorise the quadratic.

(x + 5)(x + 2) = 0

Step 3: Use the Zero Product Rule.

x + 5 = 0

or

x + 2 = 0

Step 4: Solve both equations.

x = −5

or

x = −2
The solutions are x = −5 or x = −2.

Example 2

Solve:

x² − x − 12 = 0

Find two numbers that multiply to −12 and add to −1.

−4 × 3 = −12
−4 + 3 = −1

Therefore:

(x − 4)(x + 3) = 0
x − 4 = 0

or

x + 3 = 0
x = 4

or

x = −3
The solutions are x = 4 or x = −3.

Example 3

Solve:

2x² + 10x = 0

Take out the common factor first.

2x(x + 5) = 0

Set each factor equal to zero.

2x = 0

or

x + 5 = 0
x = 0

or

x = −5
The solutions are x = 0 or x = −5.

💡 Why Does Factorisation Work?

If two quantities multiply to give zero, at least one of them must be zero.

a × b = 0

Therefore:

a = 0

or

b = 0
This is called the Zero Product Rule.

2️⃣ Method 2 — Completing the Square

Completing the square rewrites a quadratic expression in the form:

(x + p)² + q

This method can be used to solve quadratic equations and also helps us understand the shape and turning point of a quadratic graph.

Steps for completing the square when the coefficient of x² is 1

1. Move the constant to the other side if necessary.
2. Take half of the coefficient of x.
3. Square that number.
4. Add the square to both sides.
5. Write the left side as a perfect square.
6. Take the square root of both sides.
7. Remember the ± sign.

Example 4

Solve by completing the square:

x² + 6x − 7 = 0

Step 1: Move −7 to the other side.

x² + 6x = 7

Step 2: Half the coefficient of x.

6 ÷ 2 = 3

Step 3: Square 3.

3² = 9

Step 4: Add 9 to both sides.

x² + 6x + 9 = 7 + 9
(x + 3)² = 16

Step 5: Take the square root of both sides.

x + 3 = ±4

This gives two possibilities:

x + 3 = 4

or

x + 3 = −4
x = 1

or

x = −7
The solutions are x = 1 or x = −7.

3️⃣ Method 3 — The Quadratic Formula

Some quadratic equations do not factorise neatly. The quadratic formula provides a reliable method that can be used for any quadratic equation written in standard form.

x = −b ± √(b² − 4ac) 2a

The formula is used with a quadratic written as:

ax² + bx + c = 0
Letter What it represents
a The coefficient of x²
b The coefficient of x
c The constant term
The value of a can never be zero. If a = 0, the x² term disappears and the equation is no longer quadratic.

➕ What Does ± Mean?

The symbol ± means:

plus or minus

It tells us to perform two separate calculations:

Use the plus sign

−b + √(b² − 4ac)

Use the minus sign

−b − √(b² − 4ac)
That is why a quadratic equation will often have two solutions.

🧠 Steps for Using the Quadratic Formula

StudyNest Method

1. Rearrange the equation into ax² + bx + c = 0.

2. Identify a, b and c, including their signs.

3. Substitute the values into the formula using brackets.

4. Calculate the value inside the square root.

5. Use the plus sign to find the first answer.

6. Use the minus sign to find the second answer.

7. Simplify or round your answers as instructed.

📝 Worked Example: Using the Quadratic Formula

Example 5

Solve:

2x² + 3x − 2 = 0

The equation is already in the form:

ax² + bx + c = 0

Step 1: Identify a, b and c.

a = 2
b = 3
c = −2

Step 2: Substitute into the quadratic formula.

x = −3 ± √(3² − 4(2)(−2)) 2(2)

Step 3: Work out the value inside the square root.

3² − 4(2)(−2)
9 + 16 = 25

The equation now becomes:

x = −3 ± √25 4
x = −3 ± 5 4

Step 4: Use the plus sign.

x = (−3 + 5) ÷ 4
x = 2 ÷ 4
x = ½

Step 5: Use the minus sign.

x = (−3 − 5) ÷ 4
x = −8 ÷ 4
x = −2
The solutions are x = ½ or x = −2.

🔍 Another Quadratic Formula Example

Example 6

Solve:

x² − 4x − 1 = 0

Identify the coefficients carefully:

a = 1, b = −4, c = −1

Substitute into the formula:

x = −(−4) ± √((−4)² − 4(1)(−1)) 2(1)
x = 4 ± √20 2

Since √20 = 2√5:

x = 4 ± 2√5 2
The exact solutions are x = 2 + √5 or x = 2 − √5.
When an exact answer contains a square root, leave it in surd form unless the question asks for a decimal approximation.

💡 Where Does the Quadratic Formula Come From?

The quadratic formula is obtained by completing the square on the general equation:

ax² + bx + c = 0

After rearranging and completing the square, the result becomes:

x = −b ± √(b² − 4ac) 2a
You do not need to derive the formula every time you use it. However, knowing that it comes from completing the square helps connect the two methods.

🔢 The Discriminant

The expression inside the square root is called the discriminant.

b² − 4ac

The discriminant tells us how many real solutions a quadratic equation has.

Value of b² − 4ac What it means
Greater than 0 Two different real solutions
Equal to 0 One repeated real solution
Less than 0 No real solutions
A negative number has no real square root, so a negative discriminant gives no real solutions.

🌍 Real-Life Example

The height of a ball above the ground is modelled by:

h = −5t² + 20t

The ball reaches the ground when its height is zero.

−5t² + 20t = 0

Factorise:

−5t(t − 4) = 0
t = 0

or

t = 4
The ball is on the ground when it is thrown at t = 0 and when it lands again after 4 seconds.

📈 Looking Ahead

Quadratic equations can also be solved graphically.

The solutions are the x-values where the quadratic graph crosses or touches the x-axis.

We will study the complete graphical method in the Graphs topic.

⚠ Common Mistakes

Mistake 1

Trying to factorise before writing the equation equal to zero.

Mistake 2

Finding only one solution after factorising.

Mistake 3

Forgetting the ± sign when taking a square root.

Mistake 4

Using the wrong signs for a, b or c in the quadratic formula.

Mistake 5

Writing −b incorrectly when b is already negative. For example, if b = −4, then −b = −(−4) = 4.

Mistake 6

Dividing only part of the numerator by 2a. The entire numerator must be divided by 2a.

🎯 Exam Tips

Read the instruction carefully.

If the question says “solve by factorisation”, show the factors.

If it says “use the quadratic formula”, write the formula and show your substitution.

If no method is stated, choose a correct method that gives clear working.

Practice Questions

Solve the following quadratic equations.

  1. x² + 9x + 20 = 0
  2. x² − 7x + 12 = 0
  3. x² − x − 6 = 0
  4. 3x² + 6x = 0
  5. x² − 25 = 0
  6. x² + 6x − 7 = 0, using completing the square
  7. 2x² + 3x − 2 = 0, using the quadratic formula
  8. x² − 4x − 1 = 0, using the quadratic formula
  9. For x² − 6x + 9 = 0, calculate the discriminant and state the number of real solutions.
  10. For x² + 2x + 5 = 0, calculate the discriminant and state the number of real solutions.
✅ Show Answers
  1. x = −4 or x = −5
  2. x = 3 or x = 4
  3. x = 3 or x = −2
  4. x = 0 or x = −2
  5. x = 5 or x = −5
  6. x = 1 or x = −7
  7. x = ½ or x = −2
  8. x = 2 + √5 or x = 2 − √5
  9. b² − 4ac = 0, so there is one repeated real solution.
  10. b² − 4ac = −16, so there are no real solutions.

⭐ What You Can Do Now

🎉 Well Done!

You can now solve quadratic equations using several different methods.

Remember:

Write equal to zero → Choose a method → Solve both possibilities → Check

🧠 Let’s Think Together

Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.

📈 Your Progress

Lesson 15 of 22

Course Progress 94%

Exam Focus

Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.

Summary