07Dperfectsquares

= Intro to Factorising by Completing the Square =

Perfect Squares
Recall that when we factorise x² + 6x + 9 we get: math . \qquad x^2 + 6x + 9 = \Big( x + 3 \Big)^2 math

This is called a ** perfect square **.

If given the first term terms, we can find the missing constant term to produce a perfect square
 * Notice that: **
 * If you __halve__ the middle term and then __square__ it, you get the last term.
 * The value in the bracket is __half__ of the middle term.
 * Take the coefficient of x
 * Halve and then square

** Example 1 **
Add the extra value to form a perfect square math \\ . \qquad \textbf{(a)} \quad x^2 + 10x \\. \\ . \qquad \textbf{(b)} \quad x^2 - 4x \\. \\ . \qquad \textbf{(c)} \quad x^2 + x math


 * Solution:**

Adding an extra term changes the value of the original expression.
 * Problem: **

We can solve this by subtracting the same amount from the perfect squre. This keeps the expression the same value.

** Example 2 **
Form a perfect square without changing the value of the expression. math \\ . \qquad \textbf{(a)} \quad x^2 + 10x \\. \\ . \qquad \textbf{(b)} \quad x^2 - 4x \\. \\ . \qquad \textbf{(c)} \quad x^2 + x math

Notice that these are now in the form of a difference of two squares. So we could factorise. But this is the result we get from taking out x as a common factor from x² + 10x. math . \qquad x^2 + 10x = x \Big( x + 10 \Big) math
 * Solution:**

When the quadratic has three terms, it is not as easy to factorise as this.

You know how to factorise a quadratic with three terms by looking for factors of the last term. Unfortunately, only a small percentage of quadratics can be factorised in that way. For the rest, we need to use a different method.

That leads us to Factorising Quadratics by Completing the Square.

Factorising Quadratics by Completing the Square
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For more examples, go here.

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