Complete the Square When a ≠ 1
Completing the square is trickier when the x² term has a coefficient. This calculator handles quadratics where a > 1, negative, or fractional.
Complete the Square (a ≠ 1)
Factoring Out the Leading Coefficient
When a ≠ 1 in ax² + bx + c, the very first step must be factoring 'a' out of the first two terms. This leaves you with a[x² + (b/a)x] + c. You must do this before taking half of the middle term.
The Most Common Mistake
When adding the squared term inside the parentheses to complete the square, you must subtract that same value outside. However, because it's inside parentheses multiplied by 'a', the value you subtract outside must be 'a' multiplied by the squared term.
Dividing vs. Factoring
If you are solving an equation equal to 0, you can simply divide the entire equation by 'a' to get rid of it. But if you are converting a function to vertex form (y = ...), you cannot divide; you must factor 'a' out and keep it in the final equation.
Complete the Square Calculators
Specialized tools for every completing the square scenario — pick the one that matches your problem.