Question 477556
the area of a square is equal to s*s = s^2
s = 1 side of the square.
the area of the square is equal to x^2 + 6x + 9
this means that s^2 = x^2 + 6x + 9
this means that s = sqrt(x^2 + 6x + 9)
it just so happens that x^2 + 6x + 9 is a perfect square.
it is equal to (x+3)^2
this means that s^2 = (x+3)^2 which means that:
sqrt(s^2) = +/- sqrt(x+3)^2 which means that:
s = +/- (x+3)
if s = + (x+3) then:
s = x + 3
if s = -(x+3) then:
s = -x - 3
now s has to be > 0.
this means that:
x+3 > 0 which means that x > -3
this also means that:
-x-3 > 0 which means that x < -2
this means that x can be any real number except -3.
when x is equal to -3, s is equal to 0 and the area of the square is equal to 0 which means that you don't have a square of any magnitude.
the length of each side of your square is dependent on the value of x.
a graph of the equation of x^2 + 6x + 9 is shown below:
this graph shows the area of the square based on the value of x.
{{{graph(300,300,-10,10,-10,10,x^2 + 6x + 9)}}}
regardless of the value of x, this graph will be positive which means the area of the square will be positive.
this is everywhere except when x = -3.
when x = -3, the area is zero which we already indicated is not valid.
each side of your square, however, is not fixed to any one length.
the length of each side is dependent on the value of x.
a graph of the equation of y = +/- sqrt(x^2 + 6x + 9)}}} is shown below:
this graph is showing the value of the side of the square based on the value of x.
{{{graph(300,300,-10,10,-10,10,sqrt(x^2 + 6x + 9))}}}
you can see that the length of each side will always be positive.
it's low point is 0 when x = -3, which is disallowed.
you asked me to draw your square.
any square will do.
just draw a rectangle which has all sides equal to each other.
each side of this square will be equal to sqrt(x^2 + 6x + 9).
x can be any value except -3.
you can also show the value of each side of the square as follows:
s = absolute value of (x+3).
in formula terms, this looks like:
s = |x+3|
when (x+3) is positive, you get s = x + 3
when (x+3) is negative, you get s = -(x + 3) which becomes s = -x - 3.
these are the same values we calculated above.
as ugly as it might look, that's your answer.
here's a picture of your square:
the length of each side in the square is equal to |x+3|.
this means each side in the square is equal to the absolute value of (x+3).
<img src = "http://theo.x10hosting.com/problems/square_1.jpg" alt = "$$$$$"/ >
an example:
let x = -5.
each side of your square is equal to |-5+3| which is equal to |-2| which is equal to 2.
the area of your square is equal to x^2 + 6x + 9 after you replace x with (-5) to get (-5)^2 + 6*(-5) + 9 which is equal to 25 - 30 + 9 which is equal to 4.
since 2^4 = 4, the dimensions of each side and the area of the square confirm that the value of x = |-2| is good.
s is equal to 2
s^2 is equal to 4