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Quadratic Formula Calculator

What is the Quadratic Formula?

The quadratic formula is a universal method for solving any quadratic equation of the form ax² + bx + c = 0, where a ≠ 0. The formula x = (-b ± √(b² - 4ac)) / 2a gives the roots (solutions) of the equation. These roots represent the x-values where the parabola crosses the x-axis. The quadratic formula works for all cases: two distinct real roots, one repeated root, or two complex conjugate roots.

Understanding the Discriminant

The discriminant Δ = b² - 4ac is the key to understanding the nature of roots. When Δ > 0, the equation has two distinct real roots. When Δ = 0, there's exactly one real root (a repeated root). When Δ < 0, the roots are complex conjugates of the form a ± bi. The discriminant also indicates whether the parabola intersects, touches, or never crosses the x-axis.

Frequently Asked Questions

What does the discriminant tell us?

The discriminant (b² - 4ac) reveals the nature of roots: positive means two real roots, zero means one repeated root, negative means two complex roots.

Can I solve any quadratic with this formula?

Yes! The quadratic formula works for any quadratic equation ax² + bx + c = 0 where a ≠ 0, regardless of whether the roots are real or complex.

What is the vertex of a parabola?

The vertex is the turning point of the parabola. For y = ax² + bx + c, the vertex is at x = -b/(2a). It's the minimum point if a > 0, maximum if a < 0.

What are Vieta's formulas?

Vieta's formulas relate roots to coefficients: the sum of roots equals -b/a, and the product of roots equals c/a. These work even for complex roots.

How do I convert to vertex form?

Complete the square: y = a(x² + (b/a)x) + c becomes y = a(x + b/2a)² + (c - b²/4a). The vertex is (-b/2a, c - b²/4a).

What if 'a' equals zero?

If a = 0, it's no longer quadratic—it becomes a linear equation bx + c = 0, which has one solution: x = -c/b (if b ≠ 0).