Completing The Square

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The equation

cannot be factorised. However, if we add 3 to each side we get

The LHS can now be factorised to a squared expression.

If we now square root both sides of the equation we get

The right hand side could be positive or negative.

Therefore the solution to the equation will be

This method of solving quadratic equations is called Completing the Square. It involves adding (or subtracting) the necessary number to make the LHS a perfect square. The difficult part is knowing what number will give the perfect square. We will look at this in a moment, but first let's look at a few examples of completing the square.

Example 1
Add 6 to each side.
Write the LHS as a perfect square.
Square root both sides.
Take the 3 over to the RHS.

Example 2
Add 5 to each side.
Write the LHS as a perfect square.
Square root both sides.
Take the 5 over to the RHS.

Example 3
Divide everything by 2.
Add a half to both sides.
Write the LHS as a perfect square.
Square root both sides.
Take the 2 over to the RHS.

From these examples you should be able to see that once the correct number to add has been found, the method is straight forward. On the next page we will look at how to find this number.

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