Solve quadratic equations ax² + bx + c = 0 using the quadratic formula. Computes discriminant Δ, real and complex conjugate roots, vertex, and axis of symmetry.
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A quadratic equation is a second-order polynomial equation of the standard form ax² + bx + c = 0, where a ≠ 0. The solutions are called roots or zeros, geometrically representing the points where the parabola crosses the x-axis.
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \Delta = b^2 - 4acIf Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are two complex conjugate roots.
Scenario: Solve 2x² - 4x - 6 = 0
Identify coefficients: a = 2, b = -4, c = -6
Compute discriminant: Δ = (-4)² - 4(2)(-6) = 16 + 48 = 64
Apply quadratic formula: x = (4 ± √64) / (2 × 2) = (4 ± 8) / 4
x₁ = (4 + 8)/4 = 3, x₂ = (4 - 8)/4 = -1
When Δ < 0, the square root involves a negative number, meaning the parabola does not touch the real x-axis and the roots are complex conjugate numbers (p ± qi).
If a = 0, the x² term disappears, turning the quadratic equation into a first-degree linear equation (bx + c = 0).
The x-coordinate of the vertex is h = -b / (2a), and the y-coordinate is k = c - b² / (4a).
The axis of symmetry is the vertical line x = -b / (2a) that divides the parabola into two mirrored halves.
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