Enter the coefficients from a quadratic equation in standard form:
y = ax2 + bx + c
What is completing the square?
Completing the square is an algebra technique used to rewrite a quadratic expression into a form that makes the graph’s vertex easy to identify. It transforms the equation from standard form ax2 + bx + c into vertex form a(x - h)2 + k. Once the equation is in vertex form, you can instantly read the vertex as (h, k).
Why use a completing the square calculator?
Doing this by hand is important for learning, but a calculator is useful when you need speed and accuracy. It helps with:
- Checking homework and exam practice
- Converting equations quickly for graphing
- Finding vertex and axis of symmetry without extra steps
- Reviewing key algebra relationships between forms
How this calculator works
Input
You enter the values of a, b, and c from your quadratic equation.
Core formulas
The calculator computes:
- h = -b / (2a)
- k = c - b2 / (4a)
Then it writes the completed-square form as:
y = a(x - h)2 + k
Extra results provided
In addition to the transformed equation, the tool also gives:
- The vertex coordinates
- The axis of symmetry
- Whether the parabola opens upward or downward
- Real roots (if they exist)
Example
Suppose your equation is:
y = x2 + 6x + 5
After completing the square, this becomes:
y = (x + 3)2 - 4
From that form, the vertex is clearly (-3, -4), and the parabola opens upward.
Tips for students
- If a = 0, the expression is no longer quadratic.
- Keep track of signs carefully when computing h and k.
- Use vertex form to graph quickly: start at the vertex and apply vertical stretch/compression from a.
- Practice both by-hand and calculator methods for mastery.
Frequently asked questions
Is completing the square the same as factoring?
No. Factoring rewrites a quadratic as a product of binomials (when possible). Completing the square rewrites it into vertex form and works even when simple factoring is not available.
Can this method find roots?
Yes. Once in vertex form, you can set the expression equal to zero and solve. This calculator also computes roots directly from the discriminant for convenience.
What if the discriminant is negative?
Then there are no real x-intercepts. The roots are complex numbers.