Factorising harder quadratics

Description

When the coefficient of $x^2$ is greater than 1 (i.e. $a > 1$), factorising requires more care.

Method — find two numbers that multiply to $ac$ and add to $b$, then split the middle term:

Example: $6x^2 + 11x + 4$

• $a = 6$, $b = 11$, $c = 4$, so $ac = 24$
• Find two numbers that multiply to 24 and add to 11: 3 and 8
• Split: $6x^2 + 3x + 8x + 4$
• Factorise in pairs: $3x(2x + 1) + 4(2x + 1)$
• Result: $(3x + 4)(2x + 1)$

Check by expanding: $(3x + 4)(2x + 1) = 6x^2 + 3x + 8x + 4 = 6x^2 + 11x + 4$ ✓

Alternative — trial and improvement: write $(px + q)(rx + s)$ and test factor pairs of $a$ and $c$.

If factorising is not straightforward, use the quadratic formula instead.

Common error: only looking for factors of $c$ rather than factors of $ac$.

Links

Factorising quadratics Expanding quadratics Solving quadratics