# Equality of two expressionsHow to prove that x^2+6x+6 is equal to -3+(x+3)^2?

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x^2 + 6x + 6 = -3 + (x+3)^2

To prove, we will start form the left side and prove the right sides.

First we will complete the square.

==> (x^2 + 6x +6)

We will add and subtract( x's coefficient/2)^2

The coefficient of x is 6.

Then we will add and subtract (6/2)^2 = 3^2 = 9

==> x^2 + 6x +6 + 9 -9

Now we will rewrite terms.

==> x^2 + 6x +9 +6-9

==> (x^2 + 6x +9) -3 = (x+3)^2 -3 = -3 + (x+3)^2 .......q.e.d

We can prove that x^2+6x+6 is equal to -3+(x+3)^2 by opening the brackets in the expression -3+(x+3)^2 and simplifying it.

-3+(x+3)^2

=> -3 + x^2 + 9 + 6x

=> x^2 + 6x + 6

So we get x^2 + 6x + 6 starting with -3+(x+3)^2

Either we could expand the square:

(x+3)^2=x^2 + 6x + 9

We'll add -3 and we'll get:

x^2 + 6x + 9 - 3 = x^2+6x+6

Or, we could complete the square:

x^2 + 6x + 9 - 9 + 6 = (x+3)^2 - 3

Both cases, we've get the other given expression.

So, the given expressions are equivalent.