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).
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.
Practice: Factoring Quadratics
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