Quadratic Equations • Topic 3 of 4

Sum & Product of Roots

For ax² + bx + c = 0 with roots α and β, Vieta gives α + β = −b/a and αβ = c/a — and this lets you answer most "find the value of …" questions without ever solving for the roots. Want α² + β²? Use (α + β)² − 2αβ. Want 1/α + 1/β? It is (α + β)/(αβ). Want (α − β)²? It is (α + β)² − 4αβ = D/a². To build an equation from given roots, write x² − (sum)x + (product) = 0; this is the standard reverse direction CAT tests when it tells you the roots are, say, 3 and −5. A favourite CAT twist asks for a new equation whose roots are transformed (reciprocals, squares, α + 2 and β + 2) — express the new sum and product in terms of the old ones and rebuild. Keep the a in the denominator: forgetting it when a ≠ 1 is the classic slip.

✅ Solved examples

1. The roots of 2x² − 5x + 3 = 0 are α and β. Find α + β and αβ.
α + β = −b/a = 5/2; αβ = c/a = 3/2.
2. If α and β are roots of x² − 6x + 4 = 0, find α² + β².
α + β = 6, αβ = 4. α² + β² = (α + β)² − 2αβ = 36 − 8 = 28.
3. Form the quadratic whose roots are 3 and −5.
Sum = −2, product = −15. Equation: x² − (−2)x + (−15) = 0 ⇒ x² + 2x − 15 = 0.
4. If α, β are roots of x² − 7x + 10 = 0, form the equation whose roots are α + 2 and β + 2.
α + β = 7, αβ = 10. New sum = 11; new product = (α + 2)(β + 2) = αβ + 2(α + β) + 4 = 10 + 14 + 4 = 28. Equation: x² − 11x + 28 = 0.

✏️ Practice — try these, take hints as needed

1. Roots of 3x² + 12x − 6 = 0: find their sum and product.
Sum = −b/a.
Product = c/a.
Simplify −12/3 and −6/3.
Sum = −4, Product = −2
2. If α + β = 5 and αβ = 6, find α² + β².
Use (α + β)² − 2αβ.
25 − 12.
Compute.
13
3. Form a quadratic with roots 2 and 7.
Sum = 9, product = 14.
x² − (sum)x + product.
Write it out.
x² − 9x + 14 = 0
4. If α, β are roots of x² − 4x + 1 = 0, find 1/α + 1/β.
= (α + β)/(αβ).
Sum 4, product 1.
Divide.
4
5. If α, β are roots of x² − 5x + 6 = 0, form the equation with roots α² and β².
New sum = (α + β)² − 2αβ.
New product = (αβ)².
Sum 13, product 36.
x² − 13x + 36 = 0

📝 Topic test — 8 questions

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