Introduction
Many questions ask you to construct a quadratic equation when its roots (or relationships between roots) are given. This pattern converts root information into an equation quickly using the standard relations between roots and coefficients.
Mastery of this pattern allows you to form equations directly, avoid unnecessary solving, and handle transformed roots (e.g., α + k, 1/α) easily.
Pattern: Forming Equation from Roots
Pattern
Key idea: If α and β are roots of ax² + bx + c = 0 (with a ≠ 0), then a(x - α)(x - β) expands to ax² - a(α + β)x + aαβ. Use α + β = -b/a and αβ = c/a.
For a monic quadratic (a = 1) the equation with roots α and β is:
x² - (α + β)x + (αβ) = 0.
Step-by-Step Example
Question
Form the monic quadratic equation whose roots are 2 and -3.
Solution
-
Step 1: Compute sum and product of roots
α + β = 2 + (-3) = -1. αβ = 2 × (-3) = -6.
-
Step 2: Substitute into monic form
Monic equation: x² - (α + β)x + (αβ) = 0 → x² - (-1)x + (-6) = 0.
-
Step 3: Simplify
x² + x - 6 = 0.
-
Final Answer:
x² + x - 6 = 0.
-
Quick Check:
Factors (x + 3)(x - 2) ⇒ roots -3 and 2. Order doesn't matter ✅
Quick Variations
1. Non-monic equation: If you need a general a, multiply the monic form by a (choose a convenient integer to clear fractions).
2. Roots given with transformation: If roots are (α + k) and (β + k), compute new sum = (α + β) + 2k and product = αβ + k(α + β) + k².
3. Reciprocal roots: If roots are 1/α and 1/β, then sum = (α + β)/(αβ) and product = 1/(αβ). Form equation accordingly.
4. One root given, other in relation: If roots are α and mα, use sum = α(1 + m) and product = mα² to form equation; eliminate α when possible using given constraints.
Trick to Always Use
- Step 1 → Always compute sum S and product P of the requested roots first.
- Step 2 → For monic equation use x² - Sx + P = 0. If you must avoid fractions, multiply through by the least common multiple to make integer coefficients.
- Step 3 → For transformed roots, express S and P in terms of original α + β and αβ using algebraic identities (expand carefully).
Summary
Summary
Key takeaways for Forming Equation from Roots:
- Find sum S and product P of the required roots.
- Monic quadratic: x² - Sx + P = 0. For non-monic, multiply by chosen leading coefficient a.
- Use algebraic identities for transformed roots (shifts, reciprocals, scalings) to compute new S and P.
- Multiply through to clear fractions and always quick-check by expanding or factoring to verify.
Hint: Equation = x² - (sum)x + (product).
Common Mistakes: Using wrong sign for the middle term (sign error on sum).