Solving Quadratic Equations

Calculus Readiness · Axiom Academy

LESSON Solving Quadratic Equations Factoring, completing the square, and the quadratic formula — three roads to the same place: the spots where the parabola crosses the x-axis. 1. Factoring & the Zero-Product Property When a quadratic factors with nice numbers, this is the fastest route. The engine is the Zero-Product Property : a product is zero only when one of its pieces is zero. Factor the quadratic into two linear pieces, set each to zero, and read off the roots. Factor, then split into two equations Each factor = 0 is one x-intercept Solve x^2 - 5x + 6 = 0 . It factors as (x-2)(x-3)=0 , so by the zero-product property x-2=0 or x-3=0 , giving x = 2 or x = 3 — exactly the two points where the parabola touches down on the x-axis. 2. The Quadratic Formula & the Discriminant This is the universal method — it solves every quadratic, factorable or not. Both roots sit symmetrically around the parabola's axis , and the term measures how far out each crossing lands from that axis. The vertical line of symmetry through the vertex. Both roots are mirror images across it. The half-distance from the axis out to each x-intercept — one root left, one root right. Two real roots — the parabola cuts the x-axis at two distinct points. One repeated root (the vertex sits on the axis) or none (the curve misses entirely). Solve 2x^2 + 5x - 3 = 0 . Here , so the discriminant is b^2 - 4ac = 25 - 4(2)(-3) = 49 > 0 — two real roots. Then giving or x = -3 , centred on the axis .

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