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Quadratic Vertex Form

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

Quadratic Calculator

Enter your quadratic expression and get vertex form with coordinates

Quick examples:

Tips for vertex form:

  • • Enter the coefficients a, b, and c from your quadratic expression
  • • The result will be in the form a(x + h)² + k
  • • The vertex is at (-h, k)
  • • The axis of symmetry is x = -h

Master Quadratic Vertex Form with Our Advanced Tool

Our quadratic vertex form 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 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 quadratic 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 while also calculating the exact vertex coordinates.

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

Types of Quadratic Expressions

Type of QuadraticStandard FormVertex FormVertex
Simple Quadratic

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

(x+3)2−4(x + 3)^2 - 4

(−3,−4)(-3, -4)

With Leading Coefficient

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

2(x+1)2−12(x + 1)^2 - 1

(−1,−1)(-1, -1)

Negative Coefficient

x2−4x+3x^2 - 4x + 3

(x−2)2−1(x - 2)^2 - 1

(2,−1)(2, -1)

Complex Coefficients

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

3(x+1)2−13(x + 1)^2 - 1

(−1,−1)(-1, -1)

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}

(−3,−52)(-3, -\frac{5}{2})

Common Mistakes to Avoid

Wrong Vertex Sign

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).

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.

Incorrect Vertex Formula

Use h=−b2ah = -\frac{b}{2a} and k=c−b24ak = c - \frac{b^2}{4a}. Don't forget the negative sign in the hh formula!

How to Convert to Vertex Form

Converting to vertex form involves completing the square and identifying the vertex coordinates. 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

Complete the square

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

3

Rewrite as perfect square

Express as a(x + h)² + k

4

Find vertex

Vertex is at (-h, k)

Key Formulas

h=−b2ah = -\frac{b}{2a}

k=c−b24ak = 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

Vertex: (-3, -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

Vertex: (-1, -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

Vertex: (2, -1)

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Frequently Asked Questions

What is vertex form?

Vertex form is a way of writing quadratic expressions as a(x+h)2+ka(x + h)^2 + k, where (−h,k)(-h, k) is the vertex of the parabola. This form makes it easy to identify the vertex, axis of symmetry, and understand the graph's behavior.

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!

Why is vertex form useful?

Vertex form is useful because it immediately shows the vertex coordinates, makes it easy to find the axis of symmetry (x = -h), and helps determine if the parabola opens upward (a > 0) or downward (a < 0).

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. The vertex formula h=−b2ah = -\frac{b}{2a} still applies.

Is this calculator free to use?

Yes, our quadratic vertex form calculator is completely free to use with no limitations. You can convert as many quadratic expressions to vertex form as you need.

Can I use this for graphing?

Yes! Once you have the vertex form, you can easily plot the vertex and use the leading coefficient to determine the direction and width of the parabola. This is much easier than plotting from standard form.

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