Completing The Square Calculator
Complete the square for any quadratic equation with detailed step-by-step solutions
Complete the Square
Method:
Convert ax² + bx + c to a(x - h)² + k form
Vertex Form:
Frequently Asked Questions
What is completing the square?
Completing the square is a method to rewrite a quadratic equation from standard form (ax² + bx + c) to vertex form a(x - h)² + k. This reveals the vertex (h, k) and makes solving and graphing easier.
Why complete the square?
Completing the square helps solve quadratic equations, find the vertex of parabolas, derive the quadratic formula, and convert between forms. It's essential for conic sections and optimization problems.
What is the formula?
For x² + bx, add and subtract (b/2)² to create a perfect square trinomial: x² + bx + (b/2)² = (x + b/2)². This is the key step in completing the square.
What if a ≠ 1?
If a ≠ 1, first factor out 'a' from the x² and x terms only. Complete the square inside the parentheses, then distribute 'a' back through. Remember to multiply (b/2a)² by 'a' when balancing the equation.
Can you solve any quadratic this way?
Yes! Completing the square works for all quadratic equations, even those with complex or irrational solutions. It's more versatile than factoring, which only works when roots are rational.
How does it relate to the quadratic formula?
The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. Understanding completing the square helps you understand where the formula comes from and why it works.