Completing the Square Calculator – ProMathTools

Completing the Square

Transform quadratic equations into vertex form by completing the square, making it easier to find the vertex and graph parabolas.

Coefficient 'a'

Enter the coefficient of x²

Coefficient 'b'

Enter the coefficient of x

Constant 'c'

Enter the constant term

How it works

Transform quadratic equations into vertex form by completing the square, making it easier to find the vertex and graph parabolas.

The Formula

For a quadratic equation in the form ax² + bx + c = 0, completing the square involves rewriting it as a(x + h)² + k = 0, where (h, k) is the vertex.

Worked Example

Example: x² + 6x + 5 = 0
To complete the square for x² + 6x + 5 = 0: 1. Factor out 'a' (if not 1). Here, a=1. 2. Move the constant term: x² + 6x = -5 3. Take half of the x-coefficient (6/2 = 3), square it (3² = 9), and add to both sides: x² + 6x + 9 = -5 + 9 4. Factor the perfect square trinomial: (x + 3)² = 4 5. Move the constant back: (x + 3)² - 4 = 0 So, the vertex form is (x + 3)² - 4 = 0, and the vertex is (-3, -4).

Tips, Assumptions & Limitations

Remember to factor out 'a' from the x² and x terms if 'a' is not 1 before completing the square.
This method is excellent for finding the vertex of a parabola and understanding its graph.
Completing the square is also a fundamental step in deriving the quadratic formula.

FAQ

Completing the square is a powerful algebraic technique used to rewrite quadratic equations into vertex form. This form makes it easy to identify the vertex of the parabola, which is its maximum or minimum point. It's also used to solve quadratic equations and derive the quadratic formula.

The quadratic formula is a direct method to find the roots (solutions) of any quadratic equation. Completing the square is a method to transform the equation into a specific form (vertex form) from which you can also find the roots, but its primary benefit is revealing the vertex and the structure of the parabola.

Yes, our Completing the Square Calculator can be used for any quadratic equation in the standard form ax² + bx + c = 0, as long as the coefficient 'a' is not zero. It will guide you through the steps to convert it into vertex form.