07Dcompletingsquare

= Factorising by Completing the Square =

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 * Example 4 **

math . \qquad \text{Factorise : } x^2 - 4x - 12 math


 * Notice the shortcut __**DOES**__ work for this.
 * Let's pretend you could not find factors of 12 that combine to give 4
 * Completing the Square will give you the right answer


 * Solution:[[image:07Deg4b.GIF width="255" height="168" align="right"]]**

Coefficient of x is odd

 * If the coefficient of x (the middle term) is odd, use exactly the same method.
 * You will end up with fractions.
 * Use fractions and not decimals.


 * Example 5 **

math . \qquad \text{Factorise : } x^2 + 3x - 1 math


 * Solution:[[image:07Deg5b.GIF width="256" height="167" align="right"]]**

No Factors

 * Many quadratics simply cannot be factorised.
 * With this method, if you get a plus (+) between the two squared terms, you __**CANNOT**__ use the __difference__ of two squares rule.
 * Write ** "No Real Factors" ** and stop.

** Example 6 **
math . \qquad \text{Factorise : } x^2 + 2x + 7 math


 * Solution:**

** Coefficient of x² is not 1 **

 * The Completing the Square Method only works if the coefficient of x² is 1.
 * If there is a value in front of x², take it out as a common factor, even if that results in fractions.
 * Complete the square on the expression inside the bracket.

** Example 7 **
math . \qquad \text{Factorise : } 2x^2 + 2x - 3 math


 * Solution:**

** Method **

 * If the coefficient of x² is not 1, take it out the front as a common factor (even if that means some terms become fractions) and use completing the square on the expression inside the bracket.


 * Take the 2nd term (coefficient of x), halve it and then square. Add the result as a constant immediately after the 2nd term. Subtract the same amount from the end to keep the expression at the same value.


 * The first three terms should now be a perfect square. Write this as a bracket (squared). Combine the last 2 terms after the bracket.


 * If you have a plus (+) between the two terms, stop because there are ** no real factors **.


 * If you have a minus (–) between the two terms, you now have a difference of two squares.


 * Factorise using the difference of two squares rule. You may have to use surds.

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