1-7 Sum and product of the roots (Vieta)

The rules

• α + β = −b/a and αβ = c/a.
• Proof sketch: from α = (−b + √Δ)/(2a) and β = (−b − √Δ)/(2a), adding cancels √Δ and multiplying gives (b² − Δ)/(4a²) = 4ac/(4a²).

Symmetric expressions

• A symmetric expression does not change when α and β are interchanged, so it is always evaluated through α + β and αβ only.
• α² + β² = (α + β)² − 2αβ.
• α³ + β³ = (α + β)³ − 3αβ(α + β).
• (α − β)² = (α + β)² − 4αβ, so |α − β| = √Δ / |a|.
• 1/α + 1/β = (α + β)/αβ ; α/β + β/α = (α² + β²)/αβ.

Special relations between the roots

• One root twice the other: roots α and 2α → α + β = 3α, αβ = 2α²; eliminate α.
• Roots of opposite signs → αβ < 0; roots that are reciprocals → αβ = 1; roots that are negatives of each other → α + β = 0.