Sunday 12 February 2017

Completing the Square

Some quadratic equations cannot be factorised normally, so to factorise such equations, we must complete the square.

To complete the square, equations with an x squared coefficient of 1 should be written in this form:





To complete the square, the x coefficient has to be divided by two. This value would then become the value of p. The value of p has to then be squared and multiplied by -1. This value has to then be added or subtracted from the constant.
Consider this example:















In addition to this, equations with an x squared coefficient not 1 should be written in this form:





The coefficient of x squared has to be divided by the x squared coefficient as well as the x coefficient. Then the square can be completed.
An example is shown below.
















Solving equations by completing the square


To solve equations by completing the square, the value of x has to be isolated, or left on its own.
Consider this, from the previous example:

















Finding minimum and maximum points by completing the square


The value of x in the completed square which makes the equation in the bracket equal 0 is the x-coordinate of the point, and the value of y is the value of the constant. For example, the minimum point of the second equation is (-2, -13).

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