Completing the Square Calculator
Instantly convert any quadratic equation into vertex form. Get detailed, step-by-step algebraic breakdowns, vertex coordinates, and roots.
Solve Your Quadratic Equation
Enter the coefficients of your quadratic equation (ax² + bx + c = 0) to see the completed square form and a full step-by-step solution.
Equation Inputs
Enter the coefficients a, b, and c for ax² + bx + c = 0
Solution Evaluation
Vertex form and step-by-step breakdown
Quadratic Forms
Understanding the different ways to express a quadratic equation and when to use them.
| Form Name | General Structure | Best Used For |
|---|---|---|
| Standard Form | ax² + bx + c = 0 | Identifying coefficients, using the quadratic formula, finding the y-intercept (c). |
| Vertex Form | a(x – h)² + k = 0 | Graphing, identifying the vertex (h, k), and determining maximum/minimum values. |
| Factored Form | a(x – r₁)(x – r₂) = 0 | Quickly identifying the x-intercepts (roots) r₁ and r₂. |
Completing the Square FAQ
Everything you need to know about the completing the square method in algebra.
Completing the square is an algebraic technique used to rewrite a quadratic equation from standard form (ax² + bx + c = 0) into vertex form (a(x – h)² + k = 0). This makes it easier to identify the vertex of the parabola and solve for x.
We complete the square to find the maximum or minimum value of a quadratic function, to graph the parabola easily by identifying its vertex, and to derive the quadratic formula. It is also essential for solving equations that cannot be easily factored.
Yes. If the coefficient ‘a’ is not 1, you simply factor ‘a’ out of the first two terms (ax² + bx) before proceeding. You then complete the square for the expression inside the parentheses, remembering to multiply the added constant by ‘a’ when moving it outside the parentheses.
The vertex form is written as y = a(x – h)² + k, where (h, k) represents the coordinates of the vertex of the parabola, and ‘a’ determines the direction (upward or downward) and the width of the curve.