47 Solving Quadratic Equations Worksheet - Free Printable

Let's solve the problems in Equations 13, step by step, using the difference of two squares method.

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🔷 Difference of Two Squares Formula:

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

We’ll use this to factorise each expression.

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## ✔ Section A: Factorise using difference of two squares

All equations are in the form $ x^2 - a^2 = 0 $, so we can directly apply the formula.

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1) $ x^2 - 25 = 0 $

- $ 25 = 5^2 $
- So: $ x^2 - 5^2 = (x - 5)(x + 5) $
✔ Answer: $ (x - 5)(x + 5) $

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2) $ x^2 - 49 = 0 $

- $ 49 = 7^2 $
- $ x^2 - 7^2 = (x - 7)(x + 7) $
✔ Answer: $ (x - 7)(x + 7) $

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3) $ x^2 - 100 = 0 $

- $ 100 = 10^2 $
- $ x^2 - 10^2 = (x - 10)(x + 10) $
✔ Answer: $ (x - 10)(x + 10) $

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4) $ x^2 - 225 = 0 $

- $ 225 = 15^2 $
- $ x^2 - 15^2 = (x - 15)(x + 15) $
✔ Answer: $ (x - 15)(x + 15) $

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5) $ 2x^2 - 32 = 0 $

First, factor out the common number:
- $ 2x^2 - 32 = 2(x^2 - 16) $
- Now $ x^2 - 16 = x^2 - 4^2 = (x - 4)(x + 4) $
- So: $ 2(x - 4)(x + 4) $
✔ Answer: $ 2(x - 4)(x + 4) $

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6) $ 5x^2 - 45 = 0 $

Factor out 5:
- $ 5(x^2 - 9) $
- $ x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3) $
- So: $ 5(x - 3)(x + 3) $
✔ Answer: $ 5(x - 3)(x + 3) $

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7) $ 4x^2 - 144 = 0 $

Factor out 4:
- $ 4(x^2 - 36) $
- $ x^2 - 36 = x^2 - 6^2 = (x - 6)(x + 6) $
- So: $ 4(x - 6)(x + 6) $
✔ Answer: $ 4(x - 6)(x + 6) $

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8) $ 7x^2 - 567 = 0 $

Factor out 7:
- $ 7(x^2 - 81) $
- $ x^2 - 81 = x^2 - 9^2 = (x - 9)(x + 9) $
- So: $ 7(x - 9)(x + 9) $
✔ Answer: $ 7(x - 9)(x + 9) $

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## ✔ Section B: Factorise using difference of two squares

These involve variables other than $ x $, but same idea.

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1) $ 4a^2 - 9 = 0 $

- $ 4a^2 = (2a)^2 $, $ 9 = 3^2 $
- So: $ (2a)^2 - 3^2 = (2a - 3)(2a + 3) $
✔ Answer: $ (2a - 3)(2a + 3) $

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2) $ 36s^2 - 121 = 0 $

- $ 36s^2 = (6s)^2 $, $ 121 = 11^2 $
- $ (6s)^2 - 11^2 = (6s - 11)(6s + 11) $
✔ Answer: $ (6s - 11)(6s + 11) $

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3) $ 64 - p^2 = 0 $

- $ 64 = 8^2 $, so: $ 8^2 - p^2 = (8 - p)(8 + p) $
✔ Answer: $ (8 - p)(8 + p) $

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4) $ 25 - 16c^2 = 0 $

- $ 25 = 5^2 $, $ 16c^2 = (4c)^2 $
- $ 5^2 - (4c)^2 = (5 - 4c)(5 + 4c) $
✔ Answer: $ (5 - 4c)(5 + 4c) $

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5) $ x^2 - 9y^2 = 0 $

- $ x^2 - (3y)^2 = (x - 3y)(x + 3y) $
✔ Answer: $ (x - 3y)(x + 3y) $

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6) $ 63 - 7q^2 = 0 $

Factor out 7:
- $ 7(9 - q^2) $
- $ 9 - q^2 = 3^2 - q^2 = (3 - q)(3 + q) $
- So: $ 7(3 - q)(3 + q) $
✔ Answer: $ 7(3 - q)(3 + q) $

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7) $ 196 - 4y^2 = 0 $

Factor out 4:
- $ 4(49 - y^2) $
- $ 49 = 7^2 $, so $ 49 - y^2 = (7 - y)(7 + y) $
- So: $ 4(7 - y)(7 + y) $
✔ Answer: $ 4(7 - y)(7 + y) $

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8) $ 4x^2 - 121y^2 = 0 $

- $ 4x^2 = (2x)^2 $, $ 121y^2 = (11y)^2 $
- $ (2x)^2 - (11y)^2 = (2x - 11y)(2x + 11y) $
✔ Answer: $ (2x - 11y)(2x + 11y) $

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## ✔ Extension: Factorise $ 2a^3b - 8ab^3 $ using difference of two squares

This is not immediately in the form $ a^2 - b^2 $, so first factorise fully.

Step 1: Factor out the greatest common factor (GCF)

Look at:
- $ 2a^3b - 8ab^3 $

Common factors:
- Coefficients: GCF of 2 and 8 is 2
- Variables: $ a $ and $ b $ appear in both terms.
- $ a^3b $ and $ ab^3 $ → lowest powers: $ a^1 $, $ b^1 $
- So GCF is: $ 2ab $

Factor it out:
$$
2a^3b - 8ab^3 = 2ab(a^2 - 4b^2)
$$

Now look at $ a^2 - 4b^2 $ — this is a difference of two squares:
- $ a^2 - (2b)^2 = (a - 2b)(a + 2b) $

So full factorisation:
$$
2ab(a - 2b)(a + 2b)
$$

✔ Answer: $ 2ab(a - 2b)(a + 2b) $

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## 🟩 Final Answers Summary:

Section A:

1. $ (x - 5)(x + 5) $
2. $ (x - 7)(x + 7) $
3. $ (x - 10)(x + 10) $
4. $ (x - 15)(x + 15) $
5. $ 2(x - 4)(x + 4) $
6. $ 5(x - 3)(x + 3) $
7. $ 4(x - 6)(x + 6) $
8. $ 7(x - 9)(x + 9) $

Section B:

1. $ (2a - 3)(2a + 3) $
2. $ (6s - 11)(6s + 11) $
3. $ (8 - p)(8 + p) $
4. $ (5 - 4c)(5 + 4c) $
5. $ (x - 3y)(x + 3y) $
6. $ 7(3 - q)(3 + q) $
7. $ 4(7 - y)(7 + y) $
8. $ (2x - 11y)(2x + 11y) $

Extension:

$ 2ab(a - 2b)(a + 2b) $

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Let me know if you'd like these written in a printable format! 😊