# Relating the Standard and Vertex Forms: Completing the Square

```MCF 3M1
Date: _______________
Relating the Standard and Vertex Forms: Completing the Square
All quadratic functions in standard form can be written in vertex form by using a
process called Completing the Square. Both equations are equivalent.
The process works as follows:
y  4 x  16 x  5
2
y  [4 x  16 x]  5
1. Group the terms containing x with square brackets.
2. Factor the coefficient of x 2 from the first two terms.
2
y  4[ x  4 x]  5
2
y  4[ x 2  4 x  4  4]  5
y  4[( x 2  4 x  4)  4]  5
b
3. Divide the coefficient of the x term by 2 and square it.  
2
2
4. Add this term and subtract this term inside the square
y  4( x  2) 2  16  5
brackets.
y  4( x  2) 2  21
5. Group the first three terms together inside the square brackets
to make a perfect square trinomial.
6. Factor this trinomial.
7. Expand the square brackets and drop them.
8. Simplify the expression.
Example 1
Given the following equation in standard form complete the square to obtain vertex form
following the steps above
y  x2  6x  5
MCF 3M1
Example 2
Given the following equation in standard form complete the square to obtain vertex form
following the steps above
y  2 x 2  16 x  1
Example 3
Write the following quadratic function in vertex form (using fractions)
y  2 x 2  3x  7
Example 4
Write the following quadratic function in vertex form (using decimals)
y  2 x 2  3x  7
```