1-8 Forming the quadratic equation from its roots

• The equation whose roots are α, β is x² − (α + β)x + αβ = 0.
• Compute the sum, then the product, then write the rule; clear fractions by multiplying through by the L.C.M. if needed.

Roots linked to the roots of another equation

• If the new roots are 2α and 2β, then the new sum is 2(α + β) and the new product is 4αβ.
• If the new roots are α + 1 and β + 1: sum = (α + β) + 2, product = αβ + (α + β) + 1.
• Shortcut for a shift: in the original equation substitute x = y − 1 (for roots increased by 1) and expand.