Formation of Equations from Roots: Explained with Examples

If you know the roots α and β of a quadratic, the equation itself is x² − (α + β)x + αβ = 0. This is Vieta's formulas run in reverse: instead of reading root relations off a given equation, you construct the equation from its roots.

It matters in word problems and exams where roots are described rather than given — for example, "the roots are 3 more than the roots of x² − 5x + 6 = 0" — and you must produce the new equation.

Key Concepts

• Monic polynomial: a polynomial whose leading coefficient is 1, like x² − Sx + P.
• Sum and product shorthand: write S = α + β and P = αβ; then the equation is simply x² − Sx + P = 0.
• Non-uniqueness: multiplying through by any nonzero constant k gives k(x² − Sx + P) = 0, which has exactly the same roots.
• Repeated roots: if both roots equal α, the equation is (x − α)² = x² − 2αx + α² = 0.
• Cubic version: with roots α, β, γ and elementary sums S1, S2, S3, the monic cubic is x³ − S1x² + S2x − S3 = 0.

Formulas

The construction comes from multiplying the factors: (x − α)(x − β) = x² − (α + β)x + αβ. Setting this equal to zero gives the monic quadratic with exactly those roots. It is valid for real and complex α, β, and multiplying by any nonzero constant keeps the same roots.

• x² − Sx + P = 0, where S = α + β and P = αβ
• General form: k(x² − Sx + P) = 0 for any nonzero k
• Cubic: x³ − S1x² + S2x − S3 = 0

Worked Example

Form the quadratic whose roots are 2 and −5. First compute the sum: S = 2 + (−5) = −3. Then the product: P = 2 × (−5) = −10. Substitute into x² − Sx + P = 0: x² − (−3)x + (−10) = 0, which simplifies to x² + 3x − 10 = 0. Verify by substitution: for x = 2, 4 + 6 − 10 = 0; for x = −5, 25 − 15 − 10 = 0. Both roots satisfy the equation. As a cross-check, factorising gives (x − 2)(x + 5) = x² + 3x − 10, confirming the result.

Common Misconceptions

• Writing x² + Sx + P = 0. The middle term carries a minus sign: x² − Sx + P = 0. For roots 2 and −5 the equation is x² + 3x − 10 = 0 precisely because S = −3.
• "The equation is unique." 2x² + 6x − 20 = 0 has the identical roots 2 and −5; any nonzero multiple works.
• Mixing up which root goes where. Addition and multiplication are commutative, so the order of α and β never matters.

Real-World Applications

• Building quadratics with integer coefficients from integer roots in algebra exercises.
• Shifted-root problems: finding the equation whose roots are each 2 more (or k more) than a given equation's roots.
• Designing equations with desired properties, such as reciprocal or opposite roots.
• Control systems, where engineers place polynomial roots (poles) and then form the characteristic equation.

To form a quadratic from roots α and β, compute S = α + β and P = αβ and write x² − Sx + P = 0. Watch the minus sign and always verify by substitution.