Enter coefficients for a quadratic in the form ax² + bx + c = 0. This calculator rewrites it in vertex form by completing the square and shows each step.
What is completing the square?
Completing the square is a method used to rewrite a quadratic expression from standard form, ax² + bx + c, into vertex form, a(x - h)² + k. This transformation makes it much easier to identify the vertex, axis of symmetry, minimum or maximum value, and sometimes the roots of the equation.
Instead of guessing how a parabola behaves, completing the square gives you a clean structural view. It is one of the most useful techniques in algebra, precalculus, and optimization problems.
Why use a complete the square calculator?
A calculator helps you avoid arithmetic errors and instantly shows the logic behind the transformation. If you are learning, it is also a strong way to check homework and understand each algebraic step.
- Converts standard form to vertex form automatically
- Shows the vertex \((h, k)\)
- Computes the discriminant and root type
- Displays step-by-step algebra for learning and review
General formula used
For a quadratic expression \(ax^2 + bx + c\), with \(a \neq 0\):
- Vertex x-coordinate: h = -b / (2a)
- Vertex y-coordinate: k = c - b² / (4a)
- Vertex form: a(x - h)² + k
This is exactly what the calculator computes behind the scenes.
How to use this tool
Step 1: Enter coefficients
Type values for a, b, and c. You can use integers or decimals.
Step 2: Click Calculate
The calculator immediately rewrites the expression and outputs your completed-square form.
Step 3: Interpret results
Use the displayed vertex form and root information to sketch the parabola, solve equations, or verify classwork.
Practical use cases
- Graphing parabolas: Vertex form gives the turning point directly.
- Finding maxima and minima: Common in physics, business, and engineering.
- Solving quadratics: Complete the square can derive the quadratic formula itself.
- Curve fitting: Helps analyze quadratic trends in data.
Common mistakes to avoid
- Forgetting to factor out a before completing the square when \(a \neq 1\)
- Using the wrong sign in the \((x - h)\) part
- Not balancing both sides while adding and subtracting the same term
- Arithmetic errors in \(b/(2a)\) and \(b^2/(4a)\)
Example to understand quickly
Take \(x^2 + 6x + 5\):
- Half of 6 is 3, and \(3^2 = 9\)
- \(x^2 + 6x + 5 = (x^2 + 6x + 9) - 9 + 5\)
- \(= (x + 3)^2 - 4\)
So the completed-square form is (x + 3)² - 4, and the vertex is (-3, -4).
FAQ
Can this calculator handle decimals?
Yes. Enter decimal coefficients and the result will be shown in decimal form.
What if a = 0?
Then the expression is not quadratic, so completing the square is not applicable.
Does this also solve for roots?
Yes, the tool also reports whether there are two real roots, one repeated real root, or complex roots.