Quadratic Equation, Vieta's Theorem

Quadratic Equation

ax² + bx + c = 0 (a ≠ 0). Discriminant: D = b² − 4ac.

D	Roots	Formula
D > 0	two distinct	x = (−b ± √D) / 2a
D = 0	one (double root)	x = −b / 2a
D < 0	no real roots

• If b is even, take k = b/2: D₁ = k² − ac, x = (−k ± √D₁) / a.
• Incomplete equations: in ax² + c = 0, x² = −c/a; in ax² + bx = 0, x(ax + b) = 0, and the roots are 0 and −b/a.

Vieta's Theorem

If the roots are x₁ and x₂:

• x₁ + x₂ = −b/a
• x₁ · x₂ = c/a

In the reduced equation x² + px + q = 0, the sum is −p and the product is q. The sum is −b/a, that is, with the opposite sign.

Converse theorem: numbers whose sum is s and product is t are the roots of x² − sx + t = 0. Factoring: ax² + bx + c = a(x − x₁)(x − x₂).

Calculating Without Finding the Roots

x₁² + x₂² = (x₁ + x₂)² − 2x₁x₂. Similarly, 1/x₁ + 1/x₂ = (x₁ + x₂) / (x₁x₂).

1. Bring the equation to the form ax² + bx + c = 0
2. Calculate D and determine the number of roots
3. Write the sum and product using Vieta
4. Express the required expression through these two

Problems with Parameters

• The equation has one root: D = 0.
• Two distinct real roots: D > 0.
• At least one real root: D ≥ 0.

Examples

• x² − 5x + 6 = 0: the sum is 5, the product is 6, the roots are 2 and 3.
• If x² − 3x + k = 0 has one root: D = 9 − 4k = 0, k = 9/4.