Deriving the Quadratic Formula: Why Completing the Square Works

Step-by-step visual and algebraic breakdown of how ax² + bx + c = 0 yields the universal quadratic formula.

The Universal Solver for Second-Degree Polynomials

A quadratic equation is any polynomial equation where the highest exponent of the variable is 2. The standard form is ax² + bx + c = 0.

Standard Quadratic Form: ax² + bx + c = 0 (where a ≠ 0)

Step-by-Step Derivation by Completing the Square

1. Divide all terms by 'a': x² + (b/a)x + (c/a) = 0 2. Subtract (c/a): x² + (b/a)x = -c/a 3. Add (b/2a)² to both sides: x² + (b/a)x + b²/(4a²) = (b² - 4ac)/(4a²) 4. Factor left side: (x + b/2a)² = (b² - 4ac)/(4a²) 5. Take square roots and solve for x.

Universal Formula: x = (-b ± √(b² - 4ac)) / (2a)

Interpreting the Discriminant (Δ = b² - 4ac)

The expression under the radical dictates the nature and quantity of the polynomial roots.

• Δ > 0: Two distinct real roots • Δ = 0: Exactly one repeated real root • Δ < 0: Two complex conjugate roots (a ± bi)

Factoring vs Quadratic Formula

If integer factors cannot be identified within 20 seconds, immediately switch to the quadratic formula to guarantee an exact solution.

Tip: Always check if 'a' is negative; multiply the entire equation by -1 to simplify calculations.

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