Quadratic Expressions – Working with ax² + bx + c
A quadratic expression has the form ax² + bx + c, where a, b, and c are numbers and a is never zero. Unlike a quadratic equation, which sets the expression equal to zero and asks you to solve for x, a quadratic expression on its own is just a value-generating rule – you can evaluate it for any x, or rewrite it in an equivalent form, without ever needing to “solve” anything.
This distinction – expressions versus equations – mirrors what you already know from ordinary algebra: an expression like 3x + 2 has no single answer until you set it equal to something, while an equation like 3x + 2 = 11 does. The same split applies to quadratics, and getting comfortable evaluating and expanding quadratic expressions here will make the equation-solving methods on the following pages much easier to follow.
Evaluating and Expanding
To evaluate ax² + bx + c at a given x, substitute and calculate. To expand a product like (x + p)(x + q), the result is always x² + (p + q)x + pq.
2(4)² + 3(4) − 5 = 32 + 12 − 5 = 39.
x² + (3 + 5)x + (3 × 5) = x² + 8x + 15.
Real-Life Application
- Area problems: the area of an expanding rectangle is often written as a quadratic expression.
- Physics formulas: quadratic expressions describe distances under constant acceleration.
- Business modelling: profit is often modelled as a quadratic expression in terms of price.
Key Takeaways
- A quadratic expression has the form ax² + bx + c, with a ≠ 0.
- Expressions can be evaluated or expanded without solving anything.
- Expanding (x + p)(x + q) always gives x² + (p+q)x + pq.
Practice: Quadratic Expressions
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