The Discriminant Revealed

Algebra 2 · Axiom Academy

LESSON The Discriminant Revealed How the single number b^2 - 4ac tells you how many roots a quadratic has, and what kind — before you solve it. The quadratic formula solves every quadratic, and the whole story of its roots hides in one piece of it — the part beneath the square root. That part is the discriminant. It is exactly the radicand — the number under the root Take x^2 - 2x - 3 = 0 . Here b = -2 , so b^2 = (-2)^2 = 4 (not -4 — squaring a negative gives a positive), and . Watch how that 16 becomes two roots: Everything the discriminant predicts comes down to the sign of . Watch one parabola rise: as falls from positive, through zero, to negative, its roots go from two, to one, to none. Here is the reason. A parabola's turning point is its vertex, and its height relative to the x-axis is set entirely by the discriminant. For x^2 - 2x - 3 the vertex height is , four units below the x-axis. An upward parabola whose vertex is below the axis must climb back up through it — so it crosses twice . 4. The ± Collapse in the Formula Return to the formula . The is the whole drama. As shrinks to zero, and the two roots rush together; past zero, turns imaginary and the roots leave the real line as a complex-conjugate pair. , so both roots slide toward and merge into one repeated root at the axis of symmetry. becomes imaginary, so the pair lifts off the real axis to .

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