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X 2 10x-6 Put in Standard Vertex Form Calculator

Reviewed by Calculator Editorial Team

This calculator helps you convert quadratic expressions from standard form to vertex form. Vertex form is particularly useful for graphing parabolas and identifying key features like the vertex and axis of symmetry.

What is Vertex Form?

Vertex form of a quadratic equation is written as:

y = a(x - h)² + k

Where:

  • (h, k) is the vertex of the parabola
  • a determines the parabola's width and direction (upwards if a > 0, downwards if a < 0)

Converting to vertex form makes it easier to identify the parabola's maximum or minimum point and its direction.

How to Convert to Vertex Form

Step 1: Complete the Square

For the expression x² + 10x - 6:

  1. Factor the coefficient of x² (which is 1 in this case)
  2. Take half of the coefficient of x (10/2 = 5)
  3. Square this value (5² = 25)
  4. Rewrite the expression by adding and subtracting this squared value

Step 2: Rewrite as Perfect Square

The expression becomes:

x² + 10x + 25 - 25 - 6 = (x + 5)² - 31

Step 3: Final Vertex Form

The standard vertex form is:

y = (x + 5)² - 31

This shows the vertex is at (-5, -31).

Example Calculation

Let's convert x² + 10x - 6 to vertex form:

  1. Factor the coefficient of x²: 1(x² + 10x - 6)
  2. Take half of 10: 5
  3. Square 5: 25
  4. Add and subtract 25 inside the parentheses:

    5. x² + 10x + 25 - 25 - 6 = (x + 5)² - 31

  5. Final vertex form: y = (x + 5)² - 31

The vertex is at (-5, -31).

Common Mistakes

Forgetting to Factor the Coefficient

If you skip factoring the coefficient of x², you might miss the need to add and subtract the squared term. Always factor the coefficient first.

Incorrectly Calculating Half of the Coefficient

Taking half of the coefficient of x is crucial. For x² + 10x, half of 10 is 5, not 10.

Sign Errors

When adding and subtracting the squared term, be careful with signs. Forgetting to subtract the original constant can lead to incorrect results.

FAQ

Why is vertex form important?

Vertex form makes it easy to identify the vertex of the parabola, which is its maximum or minimum point. This is useful for graphing and understanding the parabola's behavior.

Can all quadratic equations be written in vertex form?

Yes, any quadratic equation can be rewritten in vertex form using the completing the square method.

What if the coefficient of x² is not 1?

If the coefficient of x² is not 1, you must factor it out first before completing the square. For example, with 2x² + 8x + 3, you would factor out 2: 2(x² + 4x) + 3.