1.1.1—Completing the square

Rewrite ax²+bx+c as a(x−h)²+k by factoring a and adding and subtracting the required square. The vertex is (h,k), and the sign of a tells whether it is a minimum or maximum.

Keep the factor outside the square: ax²+bx+c=a[x²+(b/a)x]+c. Completing the square is also the quickest route to a quadratic equation’s exact roots and range.

x²−6x+5=(x−3)²−4, so the graph has minimum −4 at x=3 and range y≥−4.

The constant changes when the square is completed; do not write (x−3)²+5, which expands to a different quadratic.