Algebra
Factoring Quadratics – Splitting an Expression into Brackets
Factoring a quadratic expression means rewriting x² + bx + c as a product of two brackets, (x + p)(x + q) – the exact reverse of the expanding you practised on the previous page. The trick is finding a pair of numbers p and q that multiply together to give c and add together to give b.
This page focuses purely on factoring the expression itself – no “= 0,” no solving for x. Once you're confident finding the right factor pair here, the next page shows how this exact same skill becomes the key technique for solving quadratic equations.
Finding the Factor Pair
To factor x² + bx + c, find two numbers p and q where p × q = c and p + q = b. The factored form is (x + p)(x + q).
Factoring as Finding Rectangle Dimensions
Factoring is expanding brackets in reverse: instead of starting with the two side lengths and finding the area, you start with the area (the four pieces below) and work out what side lengths could have produced it.
Reading the pieces backwards: the top-left x² tells you both sides start with x. The 3x on the right must come from x (the height) times 3 (a piece of the width), so the width is (x + 3). The 4x on the bottom must come from x (the width) times 4 (a piece of the height), so the height is (x + 4) — and sure enough, 3 × 4 = 12 matches the last piece exactly. Reading straight down that chain gives you the factoring process in order:
expanded expression (x² + 7x + 12) → total area → rectangle dimensions (x + 3 and x + 4) → factors (x + 3)(x + 4)
Need two numbers multiplying to 12 and adding to 7: 3 and 4 work (3 × 4 = 12, 3 + 4 = 7). Factored form: (x + 3)(x + 4).
Expanding (x + 5)(x − 3) gives x² + 2x − 15, which matches: yes, correct.
Real-Life Application
- Simplifying formulas: factored expressions are often easier to work with in engineering formulas.
- Area problems: a rectangle's dimensions can be found by factoring its area expression.
- Signal processing: factored quadratic expressions simplify certain calculations in electronics.
Key Takeaways
- Factoring rewrites x² + bx + c as (x + p)(x + q).
- Find p and q by looking for a pair that multiplies to c and adds to b.
- Factoring an expression is distinct from solving a quadratic equation, covered next.