If α and β are roots of a quadratic, how do I find α + β and αβ without solving?

For any quadratic ax² + bx + c = 0 with roots α and β, you can read the sum and product directly from the coefficients. The sum of roots α + β equals −b/a, and the product αβ equals c/a. You never need to solve the equation. For example, in 2x² − 7x + 3 = 0, here a = 2, b = −7, c = 3, so α + β = −(−7)/2 = 7/2 and αβ = 3/2. This is extremely useful for forming new equations or finding expressions like α² + β², which equals (α + β)² − 2αβ. Memorise these two results carefully, they appear in almost every quadratics question.