A community is building a square garden with a walkway around the perimeter. One side is 32 ft. The area of the inner square is equal to 75% of the total area. Write and equation for the area of the inner square. One side of the inner square is x.

I have been going back and forth between lessons and could only find some equations, but I don't know how to incorporate the numbers, I have tried many different combinations as well. I really need a push in the right direction. So The equation that is a base to use is ax^2+bx+c=0.
This is what I have
32x^2(3/4)=0
1024x(3/4)=0
768=0______
\ /768 (Finding the square root)
27.7128
So approximately the length of one side is 27.7 with what I used. But I am not entirely sure that this is correct. Someone please check this over and see what I possibly wrong.

"One side is 32 ft" is ambiguous.

Method 1
I am assuming that by "one side" you mean the side of the inside square
So the area of the square is 32^2 or 1024 ft^2

Further assumption: the walkway has the same width all around
Let the width of the walkway be x ft
So the whole thing has dimensions 32+2x by 32+2x
Area of whole thing = (32+2x)^2 = 1024 + 128x + 4x^2

75% of the whole thing = 32^2 = 1024
(3/4)(4x^2 + 128x + 1024) = 1024
3x^2 + 96x + 768 - 1024 = 0
3x^2 + 96x - 256 = 0
x = 2.475 ft, so the walkway is 2.475 ft wide

check: area of whole thing = (32 + 2*2.475)^2 = 1363.3025
and 75% of that is 1023.97... or 1024 with roundoff

Method 2.
Interpretation: The side of the original inside square is x ft
then the side of the whole thing is 32 ft by 32 ft with a total area
of 1024 ft^2
which means that the inner square is 75% of that
x^2 = .75(1024) = 768
x = √768 = appr 27.713

check: inner area = 768
outer area = 1024
is 768/1024 = .75 ? Yes it is

You pick the interpretation, state the question more clearly next time.

Thank you so much, that helps a lot!!!

I didn't even understand what the question was saying-

xD

To solve this problem, we can start by finding the total area of the square garden. Since one side of the garden is given as 32 ft, we know that the total area is (32 ft)^2 or 1024 square ft.

Next, we need to find the area of the inner square. Let's represent the length of one side of the inner square as x. The area of a square is given by the formula A = x^2.

According to the problem, the area of the inner square is equal to 75% of the total area. We can express this as an equation:

x^2 = 0.75 * 1024

Now, we can simplify the equation:

x^2 = 768

To solve for x, we can take the square root of both sides:

√(x^2) = √768

x = √768

Now, we can calculate the approximate value of x:

x ≈ 27.71 ft

So, with the information given, the length of one side of the inner square is approximately 27.71 ft.

It seems like you made a few mistakes in your calculations. You squared 32 ft instead of x, and you multiplied 32 by (3/4) instead of dividing the total area by 0.75. Additionally, the equation you mentioned (ax^2 + bx + c = 0) is not necessary for solving this problem.

I hope this helps! Let me know if you have any further questions.