When is the best time to use completing the square?

It is most useful when the quadratic equation cannot be easily factored, or when converting a quadratic function from standard form to vertex form.

Can I complete the square if 'a' is not 1?

Yes. If the coefficient 'a' is not 1, you must first divide the entire equation by 'a' before proceeding with the completing the square steps.

What happens if 'b' is 0?

If b is 0, the equation is purely quadratic (ax² + c = 0). You can solve it simply by isolating x² and taking the square root, without needing the full completing the square process.

Completing the Square Calculator — Step-by-Step Solver

Q: What is completing the square?

Completing the square is a method used to solve quadratic equations by changing the form of the equation so that the left side is a perfect square trinomial.

How Complete the Square Calculator Works

This calculator takes a standard quadratic equation in the form ax² + bx + c = 0 and performs the algebraic steps required to "complete the square." It automatically handles dividing by the coefficient 'a', moving the constant 'c', adding the square of half the middle term, and factoring the resulting perfect square trinomial.

What is Completing the Square?

Completing the square is an algebraic technique used to manipulate quadratic polynomials. It changes the equation from standard form into a perfect square form plus a constant. This technique is fundamental for:

Deriving the Quadratic Formula.
Finding the vertex (maximum or minimum point) of a parabola.
Solving quadratic equations that are difficult to factor.
Graphing quadratic functions.

How to Complete the Square

To solve ax² + bx + c = 0, follow these steps:

Move the constant term c to the right side of the equation.
Divide every term by a (if a ≠ 1) so the coefficient of x² becomes 1.
Take half of the x-coefficient (b/a), square it, and add this value to both sides of the equation.
Factor the left side as a perfect square: (x + d)².
Simplify the right side and solve for x by taking the square root.

Completing the Square when a is Not Equal to 1

If the coefficient a is not 1 (e.g., 2x² + 8x + 4 = 0), you cannot directly complete the square. You must first divide the entire equation by a. This usually introduces fractions into the problem, which this calculator handles automatically.

Completing the Square When b is 0

If b = 0, the equation looks like ax² + c = 0. In this case, there is no linear 'x' term. Completing the square is technically not needed; you can solve this directly using the Square Root Property by isolating x² (subtract c, divide by a) and taking the square root.