Factorising x² + bx + c

Statement

A quadratic expression of the form:

\[ x^2 + bx + c \]

can be factorised into two brackets:

\[ x^2 + bx + c = (x+p)(x+q) \]

where the numbers \(p\) and \(q\) satisfy the conditions:

• \(p + q = b\)
• \(p \times q = c\)

Why it works

• Expanding \((x+p)(x+q)\) gives \(x^2 + (p+q)x + pq\).
• This matches the original quadratic \(x^2 + bx + c\) when \(p+q = b\) and \(pq = c\).
• Therefore, finding the correct pair of numbers \(p, q\) allows us to factorise the quadratic into brackets.

Recipe (Steps to Factorise)

1. Write down the quadratic \(x^2 + bx + c\).
2. Find two numbers \(p, q\) such that:
   • \(p+q = b\)
   • \(p \times q = c\)
3. Write the factorised form as \((x+p)(x+q)\).
4. Check by expanding to confirm it matches the original quadratic.

Spotting it

Use this method when the coefficient of \(x^2\) is 1 (monic quadratics). If the coefficient is not 1, you need other methods such as grouping or the quadratic formula.

Common pairings

• Solving quadratic equations (\(x^2 + bx + c = 0\)).
• Sketching quadratic graphs (roots come from the factors).
• Simplifying algebraic fractions.

Mini examples

1. \(x^2 + 5x + 6 = (x+2)(x+3)\), since 2+3=5 and 2×3=6.
2. \(x^2 - x - 6 = (x-3)(x+2)\), since -3+2=-1 and -3×2=-6.

Pitfalls

• Forgetting that signs matter (both addition and multiplication conditions must hold).
• Mixing up the order — although \((x+p)(x+q)\) = \((x+q)(x+p)\).
• Trying to factorise quadratics with no integer factors (these need quadratic formula or completing the square).

Exam Strategy

• Write down factor pairs of \(c\) to test possible values quickly.
• Check sum = \(b\), product = \(c\).
• Expand your final brackets to verify correctness.

Worked Micro Example

Factorise \(x^2 + 7x + 10\).

We need two numbers that multiply to 10 and add to 7: \(5\) and \(2\).

So, \(x^2 + 7x + 10 = (x+5)(x+2)\).

Summary

To factorise \(x^2 + bx + c\), find two numbers whose product is \(c\) and sum is \(b\). This quick method works for monic quadratics and is essential for solving quadratic equations and simplifying algebraic expressions.