Solving Quadratic Equations

GRE Quantitative · Axiom Academy

LESSON Solving Quadratic Equations One parabola, one pair of roots, three different roads that all arrive at the same two points. 1. Method 1 — Factoring: Two Numbers, Two Roots Standard form is ax^2+bx+c=0 . If it factors, this is the fastest route: find two numbers that multiply to c and add to b , then read the roots straight off the factors. −4 and 2 multiply to −8, add to −2 2. Method 2 — Completing the Square: Same Roots, Different Route Not every quadratic factors nicely. Completing the square works on any quadratic by rewriting it as a perfect square, and it's the same algebra that derives the quadratic formula in Step 3. Half of -2 is -1 ; add (-1)^2=1 to both sides: x^2-2x+1=9 (x-1)^2=9 — the left side is now a single squared term x=4 and x=-2 again — completing the square just took a different algebraic path to the same two x-intercepts factoring found. 3. Method 3 — The Quadratic Formula: Works Every Time The quadratic formula packages completing the square into one formula that solves every quadratic, factorable or not — use it whenever factoring stalls. Plug in a , b , c and the roots fall out directly. Plug in the running example: , so the discriminant is b^2-4ac=4+32=36 and , giving or -2 — the same two roots a third time. The discriminant also predicts the shape: its sign tells you how many times the parabola crosses the x-axis before you solve anything. The parabola crosses the x-axis twice — exactly what the animation just traced.

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