Completing the Square Calculator

Step-by-step transformation to vertex form with live graph interpretation.

This calculator rewrites a quadratic from standard form into vertex form and shows each algebraic move. Enter coefficients from $ax^2 + bx + c$ to obtain $a(x-h)^2 + k$, together with the vertex, axis of symmetry, and a live graph of the resulting curve. Use it when you need to understand how a quadratic changes form, not simply read its final roots. For related concepts, explore Algebra Calculators.

Quick links: The transformation | Worked examples | Check your transformation

Rewrite a quadratic into vertex form using completing the square:

$$ax^2 + bx + c = a(x-h)^2 + k$$

Quick links: Live Chart | Completing the Square

Must not be 0 for a quadratic equation.

Results

Vertex: —

Quadratic summary

  • Root type: —
  • Discriminant (Δ): —
  • Standard form: —
  • Vertex form: —
  • Roots: —

Quadratic curve and vertex

What Vertex Form Makes Visible

Completing the square turns $ax^2 + bx + c$ into $a(x-h)^2 + k$. In that form, the important graph information is no longer buried in the coefficients: $(h,k)$ is the vertex, $x=h$ is the axis of symmetry, and $a$ determines whether the parabola opens upward or downward and how sharply it changes.

This tool applies to quadratics where $a \ne 0$. If $a=0$, the expression is linear rather than quadratic and should be solved as such. For a valid quadratic, vertex form can be the useful stopping point when your goal is graph interpretation, or the starting point for solving the equation by isolating a square.

The Transformation Behind Vertex Form

The method relies on adding and subtracting the same square term, so the expression changes form without changing value:

$$x^2 + px = \left(x + \frac{p}{2}\right)^2 - \left(\frac{p}{2}\right)^2$$

For a general quadratic, factor $a$ from the first two terms when needed, complete the square inside the parentheses, then combine the remaining constants. The resulting values are:

$$h = -\frac{b}{2a},\qquad k = c - \frac{b^2}{4a}$$ $$ax^2 + bx + c = a(x-h)^2 + k$$

The method card above shows this transformation for your own coefficients. Its value is not only the final vertex form, but the visible trail between the original expression and that result.

Using Vertex Form to Find Roots

When you need roots as well as the graph structure, set the vertex form equal to zero and isolate the squared term:

$$a(x-h)^2 + k = 0$$ $$(x-h)^2 = -\frac{k}{a}$$ $$x = h \pm \sqrt{-\frac{k}{a}}$$

A positive value inside the final square root gives two real roots, zero gives one repeated real root, and a negative value gives complex roots. This is the same quadratic behavior expressed through vertex form rather than through the discriminant.

Worked Transformations

A Non-Unit Leading Coefficient

Rewrite $2x^2 + 8x + 1$. Start by factoring $2$ from the quadratic and linear terms:

$$2x^2 + 8x + 1 = 2(x^2 + 4x) + 1$$ $$= 2[(x^2 + 4x + 4) - 4] + 1$$ $$= 2(x+2)^2 - 7$$

The vertex form is $2(x+2)^2 - 7$, so the vertex is $(-2,-7)$ and the axis of symmetry is $x=-2$. This is the case where factoring the leading coefficient correctly matters most.

Solving After Completing the Square

To solve $x^2 + 2x - 3 = 0$, move the constant first and then complete the square:

$$x^2 + 2x = 3$$ $$x^2 + 2x + 1 = 4$$ $$(x+1)^2 = 4$$ $$x + 1 = \pm 2$$ $$x = 1,\qquad x = -3$$

This example shows why the method is useful beyond graphing. The same transformed structure reveals the vertex and leads directly to the roots when the squared term is isolated.

Check Your Transformation Before You Use It

Most errors come from setup and sign handling rather than from the identity itself. Use this short check before relying on the result:

  1. When $a \ne 1$, factor it from the $x^2$ and $x$ terms only.
  2. After adding a square term inside the parentheses, offset its full value after accounting for the leading coefficient.
  3. Read signs carefully: $(x+q)^2$ has vertex coordinate $h=-q$.
  4. Expand the final vertex form to confirm it returns to the original expression.
  5. If you continue to solve for roots, keep both branches from the $\pm$ step.

The live graph is a useful final check. Confirm that the displayed vertex and axis match the vertex form, then use the curve to catch a sign or distribution error before reusing the result in a calculation, graph, or report.

Explore more problem-solving tools in Math Calculators.