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Completing the Square

Completing the square is when either of the following is performed:

  • x^2+ax+b is written in the form (x+\alpha)^2+\beta
  • ax^2+bx+c is written in the form \alpha(x+\beta)^2+\gamma

For the simpler case where the coefficient of x^2 is 1:

  1. Take \alpha to be half of a.
  2. Expand (x+\alpha)^2.
  3. Choose \beta so as to adjust the constant so that the original quadratic expression is obtained.

See Example 1.

For the more complicated case where the coefficient of x^2 is not 1, remove a factor of a from the original quadratic and perform the above on the inside of the brackets before expanding again in the final step. See Example 2.

Sketching Quadratics

The graph of a quadratic can easily be sketched if you think about the transformations that have been applied to the graph of y=x^2.

  • Consider the graph of y=x^2
  • The graph of say y=(x+1)^2 can be sketched by shifting the graph of y=x^2 to the left by 1. See x-transformations on the Transformations page.
  • The graph of y=3(x+1)^2 can be sketched by stretching the graph of y=(x+1)^2 about the x-axis by a factor of 3. See y-transformations on the Transformations page.
  • The graph of y=3(x+1)^2-4 can then be sketched by shifting the graph of y=3(x+1)^2 down by 4. See y-transformations again on the Transformations page.

 

Completing the Square – Board Question

Completing the Square Board Question



Example 1

Write x^2+4x+9 in form (x+\alpha)^2+\beta.


Example 2

Write 2x^2+8x-5 in the form p(x+q)^2+r.