Why do we add (b / 2a)² when completing the square?

Because expanding the binomial (x + d)² yields x² + 2dx + d². Equating 2d with b/a gives d = b/(2a). Therefore, the missing constant term needed to form a complete square is d² = (b/(2a))².

How does completing the square find the vertex of a parabola?

Completing the square converts standard form y = ax² + bx + c into vertex form y = a(x - h)² + k. The coordinates of the vertex are directly read as (h, k), where h = -b/(2a) and k = c - b²/(4a).

What happens if a quadratic equation has complex or imaginary roots?

When the right-hand side after completing the square is negative, taking the square root involves the imaginary unit i (where i = √(-1)). The solutions are complex conjugates of the form x = h ± i√|k/a|.

What if the leading coefficient a is not equal to 1?

Divide all terms by a (or factor out a) before calculating the constant term. Remember that when balancing the other side of the equation, you must multiply the added constant by a.

How do you complete the square with odd middle coefficients and fractions?

When b is an odd number (such as 5), half of b is the fraction 5/2. Squaring it produces the fraction 25/4. Add 25/4 to both sides to form (x + 5/2)².

How is the Quadratic Formula derived by completing the square?

Applying completing the square to the general equation ax² + bx + c = 0 yields (x + b/(2a))² = (b² - 4ac)/(4a²). Taking the square root of both sides gives x = (-b ± √(b² - 4ac))/(2a).

Completing the Square Calculator & Step-by-Step Vertex Form Solver

Free step-by-step Completing the Square calculator for solving quadratic equations ax² + bx + c = 0 and converting to parabolic vertex form y = a(x - h)² + k.

How to Complete the Square:

Isolate variable terms: ax² + bx = -c.
Divide or factor out leading coefficient a: x² + (b/a)x = -c/a.
Calculate the balancing constant: (b / 2a)².
Add (b / 2a)² to both sides: (x + b/(2a))² = (b² - 4ac)/(4a²).
Apply the square root property: x = -b/(2a) ± √(b² - 4ac) / (2a).