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Sum and product of roots
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If α and β are the roots of a quadratic, you can find expressions like α2 + β2 without ever finding α and β.
Sum and product
If α and β are the roots of ax2 + bx + c = 0, then
α + β = −ba
αβ = ca
For 2x2 − 5x + 1 = 0: α + β = 52 and αβ = 12.
α + β = −ba
αβ = ca
For 2x2 − 5x + 1 = 0: α + β = 52 and αβ = 12.
Useful rearrangements
α2 + β2 = (α + β)2 − 2αβ
1α + 1β = (α + β)αβ
(α − β)2 = (α + β)2 − 4αβ
α3 + β3 = (α + β)3 − 3αβ(α + β)
1α + 1β = (α + β)αβ
(α − β)2 = (α + β)2 − 4αβ
α3 + β3 = (α + β)3 − 3αβ(α + β)
Example
x2 − 4x + 1 = 0: α + β = 4, αβ = 1
α2 + β2 = 16 − 2 = 14
1α + 1β = 41 = 4
α2 + β2 = 16 − 2 = 14
1α + 1β = 41 = 4
α and β are the roots of 2x2 − 5x + 1 = 0. Find α2 + β2.
- α + β = 52, αβ = 12
- α2 + β2 = (52)2 − 2 × 12
- = 254 − 1
Answer: 214
α + β = −ba and αβ = ca. Rewrite every expression in terms of α + β and αβ.
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