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Completing The Square and Put Into Vertex Form Calculator

Reviewed by Calculator Editorial Team

Completing the square is a method used to rewrite quadratic equations in vertex form, which makes it easier to identify the vertex of the parabola represented by the equation. This technique is essential in algebra and calculus for analyzing quadratic functions.

What is Completing the Square?

Completing the square is a technique used to rewrite quadratic equations in the form of a perfect square trinomial. This process helps to identify the vertex of the parabola represented by the quadratic equation. The standard form of a quadratic equation is:

y = ax² + bx + c

By completing the square, we can rewrite this equation in vertex form:

y = a(x - h)² + k

where (h, k) represents the vertex of the parabola.

Vertex Form Equation

The vertex form of a quadratic equation is particularly useful because it clearly identifies the vertex of the parabola. The general vertex form is:

y = a(x - h)² + k

Here, (h, k) is the vertex of the parabola, and 'a' determines the parabola's direction and width.

If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.

How to Complete the Square

To complete the square, follow these steps:

  1. Start with the standard form equation: y = ax² + bx + c.
  2. Factor out the coefficient of x² from the first two terms: y = a(x² + (b/a)x) + c.
  3. Take half of the coefficient of x, square it, and add and subtract this value inside the parentheses: y = a(x² + (b/a)x + (b/2a)² - (b/2a)²) + c.
  4. Rewrite the perfect square trinomial as a squared binomial: y = a[(x + b/2a)² - (b/2a)²] + c.
  5. Distribute 'a' and combine like terms to get the vertex form: y = a(x + b/2a)² + (c - ab²/4a²).

This process transforms the quadratic equation into vertex form, making it easier to identify the vertex and analyze the parabola.

Example Calculation

Let's complete the square for the equation y = 2x² + 8x + 3.

  1. Factor out the coefficient of x²: y = 2(x² + 4x) + 3.
  2. Take half of the coefficient of x (which is 4), square it (16), and add and subtract it inside the parentheses: y = 2(x² + 4x + 16 - 16) + 3.
  3. Rewrite the perfect square trinomial: y = 2[(x + 2)² - 16] + 3.
  4. Distribute '2' and combine like terms: y = 2(x + 2)² - 32 + 3 = 2(x + 2)² - 29.

The vertex form of the equation is y = 2(x + 2)² - 29, with the vertex at (-2, -29).

FAQ

What is the purpose of completing the square?
Completing the square is used to rewrite quadratic equations in vertex form, which makes it easier to identify the vertex of the parabola and analyze the quadratic function.
How do I know if I've completed the square correctly?
You can verify your work by expanding the vertex form equation and checking if it matches the original standard form equation.
Can completing the square be used for any quadratic equation?
Yes, completing the square can be applied to any quadratic equation in the form y = ax² + bx + c.
What is the vertex form of a quadratic equation?
The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.
How does completing the square help in graphing quadratic functions?
Completing the square helps identify the vertex and axis of symmetry, making it easier to sketch the graph of the quadratic function.