ACT-Style: Quadratics with No Real Solutions

ACT Math · Axiom Academy

EXAMPLE ACT-Style: Quadratics with No Real Solutions Using the discriminant to identify when a quadratic has zero real solutions How many real solutions does the equation 2x^2 - 4x + 5 = 0 have? The discriminant tells you the number of real solutions before you ever solve for x — no calculator required. Discriminant determines the nature of a quadratic's solutions. means zero real solutions (two complex conjugate solutions) — never assume a quadratic automatically has two real roots. Watch the sign under the radical: is not a real number, so plugging ahead and computing a decimal anyway is the exact trap the ACT sets — it produces two fake "real" answers that don't actually satisfy the equation. Result: 2x^2-4x+5=0 has , so it has zero real solutions — answer choice A. Every "how many real solutions" ACT question reduces to one check: the sign of b^2-4ac . Compute it, read its sign, and you have the answer — no further algebra needed.

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