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Completing the Square

Convert quadratic expressions to vertex form effortlessly with our step-by-step calculator. Get instant completed square form and detailed explanations for quadratic equations.

Quadratic Calculator

Enter your quadratic expression and get vertex form with steps

Quick examples:

Tips for completing the square:

  • • Enter the coefficients a, b, and c from your quadratic expression
  • • The result will be in the form a(x + h)² + k
  • • This form is useful for finding the vertex and solving equations
  • • The vertex is at (-h, k) when a = 1

Master Completing the Square with Our Advanced Tool

Our completing the square calculator is designed to help students, teachers, and professionals convert quadratic expressions to vertex form efficiently. Whether you're working on algebra homework, preparing for SAT exams, or tackling university mathematics courses, this tool provides comprehensive step-by-step solutions that enhance your understanding of quadratic transformations.

The quadratic vertex form calculator handles expressions in standard form ax2+bx+cax^2 + bx + c and converts them to vertex form a(x+h)2+ka(x + h)^2 + k. This completing square calculator is particularly useful for SAT preparation, where understanding the vertex form is crucial for graphing and solving problems. Our algebra calculator provides detailed steps showing how to factor out the leading coefficient, complete the square, and simplify to vertex form.

Perfect for high school algebra students learning quadratic functions, college studentsin calculus prerequisites, and professionals who need quick mathematical solutions. The completing the square tool provides not just the vertex form, but detailed explanations that help you understand the underlying concepts and improve your problem-solving skills.

Types of Quadratic Expressions

Type of QuadraticStandard FormVertex FormDifficulty Level
Simple Quadratic

x2+6x+5x^2 + 6x + 5

(x+3)2−4(x + 3)^2 - 4

Beginner
With Leading Coefficient

2x2+4x+12x^2 + 4x + 1

2(x+1)2−12(x + 1)^2 - 1

Intermediate
Negative Coefficient

x2−4x+3x^2 - 4x + 3

(x−2)2−1(x - 2)^2 - 1

Beginner
Fractional Coefficients

12x2+3x+2\frac{1}{2}x^2 + 3x + 2

12(x+3)2−52\frac{1}{2}(x + 3)^2 - \frac{5}{2}

Advanced
Complex Coefficients

3x2+6x+23x^2 + 6x + 2

3(x+1)2−13(x + 1)^2 - 1

Intermediate

Common Mistakes to Avoid

Forgetting to Factor Out a

When a≠1a \neq 1, you must factor out aa first. For 2x2+4x+12x^2 + 4x + 1, start with 2(x2+2x)+12(x^2 + 2x) + 1, not directly completing the square.

Wrong Sign in Vertex

Remember: (x+h)2(x + h)^2 means the vertex is at (−h,k)(-h, k), not (h,k)(h, k). For (x+3)2−4(x + 3)^2 - 4, the vertex is at (−3,−4)(-3, -4).

Not Distributing Back

After completing the square, don't forget to distribute the leading coefficient back. For 2(x+1)2−12(x + 1)^2 - 1, the 22 must be distributed.

How to Complete the Square

Completing the square is a technique used to convert quadratic expressions from standard form to vertex form. The key steps are:

ax2+bx+c→a(x+h)2+kax^2 + bx + c \rightarrow a(x + h)^2 + k

The vertex form makes it easy to identify the vertex at (−h,k)(-h, k) and understand the graph's behavior.

Step-by-Step Method

1

Factor out a

If a ≠ 1, factor it out from the first two terms

2

Add and subtract

Add and subtract (b/2a)² inside the parentheses

3

Complete the square

Rewrite as a perfect square trinomial

4

Simplify

Combine constant terms and distribute if needed

Key Formula

h=b2a,k=c−b24ah = \frac{b}{2a}, \quad k = c - \frac{b^2}{4a}

Examples

Simple Quadratic

x2+6x+5x² + 6x + 5

Solution:

  1. Add and subtract 9: x2+6x+9−9+5x^2 + 6x + 9 - 9 + 5

  2. Complete the square: (x+3)2−4(x + 3)^2 - 4

(x+3)2−4(x + 3)² - 4

With Leading Coefficient

2x2+4x+12x² + 4x + 1

Solution:

  1. Factor out 2: 2(x2+2x)+12(x^2 + 2x) + 1

  2. Add and subtract 1: 2(x2+2x+1−1)+12(x^2 + 2x + 1 - 1) + 1

  3. Complete: 2(x+1)2−12(x + 1)^2 - 1

2(x+1)2−12(x + 1)² - 1

Negative Coefficient

x2−4x+3x² - 4x + 3

Solution:

  1. Add and subtract 4: x2−4x+4−4+3x^2 - 4x + 4 - 4 + 3

  2. Complete the square: (x−2)2−1(x - 2)^2 - 1

(x−2)2−1(x - 2)² - 1

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Frequently Asked Questions

What is completing the square?

Completing the square is a technique used to convert quadratic expressions from standard form ax2+bx+cax^2 + bx + c to vertex form a(x+h)2+ka(x + h)^2 + k. This makes it easier to find the vertex and understand the graph of the quadratic function.

Why is vertex form useful?

Vertex form makes it easy to identify the vertex at (−h,k)(-h, k), determine the axis of symmetry, and understand how the parabola opens (upward if a>0a > 0, downward if a<0a < 0). It's also useful for solving quadratic equations.

How do I find the vertex from vertex form?

In vertex form a(x+h)2+ka(x + h)^2 + k, the vertex is at (−h,k)(-h, k). For example, in (x+3)2−4(x + 3)^2 - 4, the vertex is at (−3,−4)(-3, -4). Remember to change the sign of hh!

What if the leading coefficient is not 1?

When a≠1a \neq 1, you must first factor out aa from the first two terms, then complete the square inside the parentheses, and finally distribute aa back. This is shown in our step-by-step solutions.

Is this calculator free to use?

Yes, our completing the square calculator is completely free to use with no limitations. You can complete the square for as many quadratic expressions as you need.

Can I use this for solving quadratic equations?

Yes! After completing the square, you can easily solve quadratic equations by setting the expression equal to zero and using square roots. This is often easier than using the quadratic formula.

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Last updated: 24/08/2025 — Written by the AskMathAI team