Factoring Quadratic Equations - Free Printable

It looks like you've uploaded an image of a completed worksheet titled *"Factorising Quadratic Equations – Sheet 9 Answers"*, which contains solutions to 16 quadratic equations. Since the image already includes the correct answers, I’ll explain how to solve one of these problems step-by-step (for example, problem #1), and then summarize the general method for factorizing quadratics.

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🔍 Example: Problem #1

Equation:
$$
2x^2 + 5x + 2 = 0
$$

#### Step 1: Identify the coefficients
This is in the form $ ax^2 + bx + c = 0 $, where:
- $ a = 2 $
- $ b = 5 $
- $ c = 2 $

We need to find two numbers that:
- Multiply to $ a \times c = 2 \times 2 = 4 $
- Add to $ b = 5 $

Let’s list factor pairs of 4:
- $ 1 \times 4 = 4 $ → $ 1 + 4 = 5 $ ✔

Perfect! So we use 1 and 4.

#### Step 2: Split the middle term
Rewrite the equation by splitting $ 5x $ into $ 4x + x $:
$$
2x^2 + 4x + x + 2 = 0
$$

#### Step 3: Factor by grouping
Group terms:
$$
(2x^2 + 4x) + (x + 2) = 0
$$

Factor each group:
$$
2x(x + 2) + 1(x + 2) = 0
$$

Now factor out the common binomial $ (x + 2) $:
$$
(2x + 1)(x + 2) = 0
$$

✔ This matches the answer given in the sheet!

#### Step 4: Solve for $ x $
Set each factor equal to zero:
- $ 2x + 1 = 0 $ → $ x = -\frac{1}{2} $
- $ x + 2 = 0 $ → $ x = -2 $

So the solutions are:
$$
x = -\frac{1}{2} \quad \text{or} \quad x = -2
$$

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🧠 General Method for Factorizing Quadratics (when $ a \neq 1 $)

For a quadratic equation $ ax^2 + bx + c = 0 $:

1. Multiply $ a \times c $
2. Find two numbers that multiply to $ ac $ and add to $ b $
3. Split the middle term using those numbers
4. Factor by grouping
5. Solve by setting each factor equal to zero

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💡 Tips & Tricks

- If $ a = 1 $, it's easier: just find two numbers that multiply to $ c $ and add to $ b $
- Always check your factoring by expanding to verify
- Use the AC method (as above) when $ a \neq 1 $
- For practice, try solving another one — say, #2: $ 2x^2 + 4x + 2 = 0 $

Let me know if you'd like me to walk through another problem or help you understand any specific one!