How the quadratic formula works
For ax² + bx + c = 0 with a ≠ 0, the quadratic formula gives both roots directly:
x = [−b ± √(b² − 4ac)] ÷ 2a
The expression under the square root, the discriminant D = b² − 4ac, decides everything before you even take the root. D > 0 means two distinct real roots (the parabola crosses the x-axis twice), D = 0 means one repeated real root (the parabola just touches the axis at its vertex), and D < 0 means no real roots at all (the parabola never reaches the axis, so the roots are complex). This one number is the whole story of a quadratic.
Worked example
Solve x² − 5x + 6 = 0: here a = 1, b = −5, c = 6, so D = 25 − 24 = 1 — two real roots. Then x = (5 ± 1) ÷ 2, giving x₁ = 3 and x₂ = 2. (Factoring confirms it: (x − 3)(x − 2) = x² − 5x + 6.) The vertex sits at x = −b ÷ 2a = 2.5, with y = 2.5² − 5(2.5) + 6 = −0.25, the lowest point of the curve.
Completing the square and why the vertex matters
The same roots emerge by rewriting the equation in vertex form, y = a(x − h)² + k, where h = −b ÷ 2a and k is the value you get by substituting h. Because a square is never negative, the minimum (if a > 0) or maximum (if a < 0) of any parabola is exactly that vertex value k. So D < 0 is really the statement that |k| > 0, and a > 0 < 0 is the statement that the vertex sits above or below the axis.
Practical uses: the vertex gives the maximum revenue or minimum cost of a pricing quadratic; the roots give break-even volumes; and the discriminant is the quickest way to check whether a word problem has a sensible real answer before you start solving.