Vieta's Formulas for Quadratic Equations and Root Transformations
In higher algebra, Vieta's formulas establish an elegant structural connection between the roots of a polynomial and its coefficients. For a quadratic equation ax² + bx + c = 0 with roots L and M, the sum of the roots is given by L + M = -b/a, and their product is given by L · M = c/a. These fundamental identities enable mathematicians to evaluate symmetric algebraic expressions—such as L² + M², 1/L + 1/M, and (L - M)²—without solving for the individual roots. Furthermore, they provide an algorithmic method to formulate a new quadratic equation whose roots are algebraic functions of the original roots. In this lesson, students explore symmetric polynomial identities and their applications.
1. Vieta's Fundamental Relations
If L and M are the roots of the quadratic equation ax² + bx + c = 0 (where a ≠ 0), then:
Sum of the Roots
L + M = -b / a
Product of the Roots
L · M = c / a
Special Conditions & Direct Inferences
- One root is the additive inverse of the other: L + M = 0 ⟹ -b/a = 0 ⟹ b = 0.
- One root is the multiplicative inverse (reciprocal) of the other: L · M = 1 ⟹ c/a = 1 ⟹ a = c.
2. Essential Symmetric Algebraic Identities
Every symmetric function of the roots can be expressed entirely in terms of the sum (L + M) and product (LM):
| Symmetric Expression | Identity in terms of (L + M) and (LM) | Derivation & Algebraic Justification |
|---|---|---|
| L² + M² | (L + M)² - 2LM | Since (L + M)² = L² + 2LM + M², subtract 2LM. |
| (L - M)² | (L + M)² - 4LM = (b² - 4ac) / a² | Expanding yields L² - 2LM + M² = (L² + M²) - 2LM. |
| |L - M| (Absolute Difference) | √(b² - 4ac) / |a| | The absolute distance between the two roots. |
| 1/L + 1/M | (L + M) / (LM) = -b / c | Common denominator: (M + L) / (LM). |
| L³ + M³ | (L + M)[(L + M)² - 3LM] | Factoring sum of cubes: (L + M)(L² - LM + M²). |
3. Forming a Quadratic Equation Given Its Roots
To construct a quadratic equation whose roots are α and β, apply the universal canonical form:
x² - (Sum of Roots)x + (Product of Roots) = 0
x² - (α + β)x + (α · β) = 0
Worked Analytical Problems
Problem: Forming a Transformed Quadratic Equation
If L and M are the roots of x² - 5x + 3 = 0, form the quadratic equation whose roots are (L + 1) and (M + 1).
1. From original equation: L + M = -(-5)/1 = 5, and LM = 3/1 = 3.
2. Calculate Sum of new roots: (L + 1) + (M + 1) = (L + M) + 2 = 5 + 2 = 7.
3. Calculate Product of new roots: (L + 1)(M + 1) = LM + (L + M) + 1 = 3 + 5 + 1 = 9.
4. Form the equation: x² - (Sum)x + (Product) = 0 ⟹ x² - 7x + 9 = 0.
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