Definitions

• Factorising: the opposite of expanding; it means "putting into brackets".
• Factors: parts of a number that divide exactly into it. For example, 5 divides into 10.
• Quadratic: an expression with degree 2.
• Degree: the highest power of a variable in an expression.

Note that this is only for "higher" students.

Factorising \(ax^2 + bx + c\)

Now, here comes a big issue. You cannot factorise these quadratics like you do when \(a=1\).

Have no fear! You've come too far now to give up. These quadratics require you to apply everything you've learned about expressions.

AC Method

1. Check if you can divide the whole expression to simplify.
2. Multiply \(a\) by \(c\).
3. Find the factors of \(ac\).
4. Rewrite the expression as \(ax^2 + px + qx + c\) where \(p + q = b\).
5. Factorise in parts.

Example 1

Factorise \(4x^2 + 25x + 36\)

1. Can't simplify any further by dividing.
2. Calculate \(a \times c = 4 \times 36 = 144\).
3. Factors of 144: (1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16), and (12, 12). Find a pair that sums to \(b = 25\).
4. \((9,16)\) is the pair. Now write as \(4x^2 + 16x + 9x + 36\). Make sure you put the correct numbers next to each other. For example, you could write \(4x^2 + 9x + 16x + 36\), but that won't work because 4 and 9 do not have a common factor greater than 1.
5. Factorise in parts by finding the highest common factor (HCF). Factorise in pairs: - \(4x^2 + 16x = 4x(x + 4)\) - \(9x + 36 = 9(x + 4)\)
6. Rewrite as \(4x(x + 4) + 9(x + 4)\). Finally, factor out the common binomial: \((4x + 9)(x + 4)\).

Example 2

Factorise \(36x^2 + 36x + 9\)

1. Can divide by 9 to get \(4x^2 + 4x + 1\) — this saves time because the numbers are smaller and easier.
2. Calculate \(a \times c = 4 \times 1 = 4\).
3. Factors of 4: (1, 4), (2, 2). Find a pair that sums to the new \(b = 4\).
4. \((2, 2)\) is the pair. Now write as \(4x^2 + 2x + 2x + 1\).
5. Factorise in parts: - \(4x^2 + 2x = 2x(2x + 1)\) - \(2x + 1 = 1(2x + 1)\)
6. Rewrite as \(2x(2x + 1) + 1(2x + 1) = (2x + 1)(2x + 1) = (2x + 1)^2\).
7. Finally, add back the 9 you factored out at the start: \[ 9(2x + 1)^2 \] Remember this rule: whenever you divide an expression, it doesn't disappear — it becomes a factor outside the brackets.