Quick Answer: The Quadratic Formula and Root Types
Every quadratic equation in standard form ax² + bx + c = 0 (where a ≠ 0) can be solved using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) The expression under the square root, Δ = b² - 4ac, is called the Discriminant. It reveals the exact nature of the solutions without having to solve the entire equation: • If Δ > 0: Two distinct real roots exist. • If Δ = 0: Exactly one repeated real root exists (x = -b / 2a). • If Δ < 0: No real roots exist; the solutions are a pair of complex conjugate roots (x = u ± vi). Solve equations with full step-by-step working and parabola vertex coordinates on our [Quadratic Equation Solver](/quadratic-equation-solver).
Worked Example 1: Two Distinct Real Roots (Δ > 0)
Solve: 2x² - 4x - 6 = 0 • Coefficients: a = 2, b = -4, c = -6 Step 1: Compute the Discriminant (Δ) Δ = b² - 4ac = (-4)² - 4(2)(-6) = 16 - (-48) = 16 + 48 = 64 Since 64 > 0, the equation has two distinct real roots. Step 2: Apply the Quadratic Formula √Δ = √64 = 8 x = [-(-4) ± 8] / (2 × 2) = (4 ± 8) / 4 • Root 1: x₁ = (4 + 8) / 4 = 12 / 4 = 3 • Root 2: x₂ = (4 - 8) / 4 = -4 / 4 = -1 Roots are x = 3 and x = -1.
Worked Example 2: Complex Conjugate Roots (Δ < 0)
Solve: x² + 2x + 5 = 0 • Coefficients: a = 1, b = 2, c = 5 Step 1: Compute Discriminant Δ = (2)² - 4(1)(5) = 4 - 20 = -16 Since Δ < 0, roots are complex. Step 2: Apply the Formula with imaginary unit i (where i = √-1) √-16 = 4i x = [-2 ± 4i] / (2 × 1) = -1 ± 2i Roots are x = -1 + 2i and x = -1 - 2i.
Finding the Vertex and Axis of Symmetry
Every quadratic equation corresponds geometrically to a parabola y = ax² + bx + c: • Axis of Symmetry: The vertical line passing through the turning point: x = -b / (2a). • Vertex Coordinates: (h, k) where h = -b / (2a) and k = c - (b² / 4a). • Opening Direction: If a > 0, the parabola opens upward with a minimum value at the vertex. If a < 0, the parabola opens downward with a maximum value at the vertex.
What Happens When a = 0?
If coefficient a = 0, the equation reduces to bx + c = 0, which is no longer quadratic—it is a linear equation! The solution is simply x = -c / b (assuming b ≠ 0). Our solver automatically recognizes when a = 0 and solves it as a linear equation with explicit notice.