Completing the Square

Pre-Calculus · Axiom Academy

EXAMPLE Completing the Square: 3x^2 - 12x + 7 to Vertex Form Completing the square when the leading coefficient isn't 1 — then reading the vertex. Rewrite f(x) = 3x^2 - 12x + 7 in vertex form a(x-h)^2 + k by completing the square, then state the vertex and the minimum value. Nicely done — you completed the square with a leading coefficient other than 1 and landed on vertex form. Factor a first: completing the square needs a leading coefficient of 1 , so pull a=3 out of the x -terms before anything else. Half, then square: inside x^2 - 4x , half of -4 is -2 and (-2)^2 = 4 — that is the number you add and subtract. Distribution scales the correction: the outer 3 turns the inside -4 into -12 , so the constant is -12 + 7 = -5 . Result: 3x^2 - 12x + 7 = 3(x-2)^2 - 5 , with vertex (2, -5) — a minimum, since . Vertex form makes the turning point and the minimum value of the parabola immediate — no graphing required.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.