Take a look at the expression below:
x2 + 4x + 3
To complete the square you have to focus on the number before the x or the (x coefficient).
In this case, this number is 4.
To complete the square you take that number (x coefficent) and halve it, then square it.
Therefore: 4/2 = 2 -----> 22 = 4
We then add this number after the x and also minus it after the last number (constant):
x2 + 4x + 4 + 3 - 4
Completing the square is about being able to factorise, which is why this expression can now be factorised:
(x2 + 4x + 4) + 3 - 4
The brackets factorise to --> (x + 2)2 whilst the digits outside the brackets equate to -1
Therefore, our completed expression would now look like: (x + 2)2 - 1
The reason this is useful is because if our original expression was an equation it would look like this:
x2 + 4x + 3 = 0
Therefore, our new equation would look like this:
(x+2)2 - 1 = 0
With our original equation the only way we could solve it is by using the quadratic formula.
But with our new factorised equation we can solve for x by quick algebra manipulation:
--> (x+2)2 - 1 = 0
--> (x+2)2 = 1
--> (x+2) = sqrt(1)
--> x + 2 = 1
--> x = 1 - 2
Therefore: x = -1
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