Sum and Product of Quadratic Roots

Quick answer: If α and β are roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. These follow by expanding a(x − α)(x − β).

Worked example

Find the sum and product of roots of 2x² − 7x + 3 = 0.

1. Identify a = 2, b = −7, c = 3.
2. Sum = −(−7)/2 = 7/2.
3. Product = 3/2.
4. Factorisation gives (2x − 1)(x − 3), so the roots are 1/2 and 3.

Check the result

1/2 + 3 = 7/2 and (1/2)(3) = 3/2.

Common mistake to avoid

The minus sign belongs in the sum formula, not the product formula. Include the coefficient a rather than assuming the quadratic is monic.

Try it yourself

1. Find the sum and product for 3x² + 6x − 9. 2. Form a monic quadratic with root sum 5 and product 6.

Answers and reasoning

1. Sum −2, product −3. 2. x² − 5x + 6; its roots are 2 and 3.

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