How do I complete the square

Take a look at the expression below:

x+ 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 -----> 2= 4

We then add this number after the x and also minus it after the last number (constant):

x+ 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:

x+ 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

Answered by Anant M. Maths tutor

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