General Steps for Completing the Square
Start
With and equation of the form:
\$ax^2 + bx + c = 0\$
Step 1
Put parentheses around the x terms:
\$(ax^2 + bx) + c = 0\$
Step 2
Factor out the a term:
\$a(x^2 + b/ax) + c = 0\$
Step 3
Inside the parentheses, add \$(1/2*b/a)^2\$:
\$a(x^2 + b/ax + (b/(2a))^2) + c = 0\$
Step 4
Outside the parentheses, subtract \$b^2/(4a)\$:
\$a(x^2 + b/ax + (b/(2a))^2) + c - b^2/(4a) = 0\$
| This balances Step 3 by subtracting the same value as was added. |
Step 5
Factor the quadratic term within the parentheses:
\$a(x + b/(2a))^2 + c - b^2/(4a) = 0\$
Example of Completing the Square
Start
\$6x^2 + 18x - 25 = 0\$
Step 1
\$(6x^2 + 18x) - 25 = 0\$
Step 2
\$6(x^2 + 18/6x) - 25 = 0\$
Step 3
\$6(x^2 + 3x + (18/(2*6))^2) - 25 = 0\$
Step 4
\$6(x^2 + 3x + (3/2)^2) - 25 - 18^2/(4*6)= 0\$
Step 5
\$6(x + 3/2)^2 - 25 - 324/24 = 0\$
Simplify
\$6(x + 3/2)^2 - 50/2 - 27/2 = 0\$
\$6(x + 3/2)^2 - 77/2 = 0\$