[tex]a-the\ side\ of\ square\\a^2-the\ area\ of\ square\\\\a+3-the\ one\ side\ of\ rectangle\\a-2-the\ second\ side\ of\ rectangle\\(a+3)(a-2)-the\ area\ of\ rectangle\\\\\text{The equation:}\\\\(a+3)(a-2)=a^2+30\qquad\text{use distributive property}\\\\(a)(a)+(a)(-2)+(3)(a)+(3)(-2)=a^2+30\\\\a^2-2a+3a-6=a^2+30\qquad\text{subtract}\ a^2\ \text{from both sides}\\\\a-6=30\qquad\text{add 6 to both sides}\\\\\boxed{a=36}\\\\Answer:\ \boxed{36\ cm}[/tex]
By writing and solving a system of equations, we want to find the side of the given square. We will see that the side length of the square is S = 5.52cm
Let's see how to solve the problem:
So we know that the side of the square, S, is 3cm smaller than one side of a rectangle, L, and 2 cm greater than the other side of the rectangle, W.
So we can write this as:
We also know that the area of the rectangle is 30cm^2, remember that the area is just the product between the measures of both sides, so we have:
L*W = 30cm^2
We can write this as:
L = (30cm^2)/W
And replace it on the first equation to get the system of equations:
So now we have two equations and two variables.
We want to find the value of S, so we can isolate W in one of the equations, if we isolate W in the second one we get:
W = S - 2cm
Now we can replace this on the first equation to get:
S = (30cm^2)/(S - 2cm) - 3cm
Now let's multiply both sides by (S - 2cm)
S*(S - 2cm) = 30cm^2 - 3cm*(S - 2cm)
S^2 - 2cm*S = 30cm^2 - 3cm*S + 6cm^2
Moving all the terms to the same side we get:
S^2 + 1cm*S - 36cm^2 = 0
This is just a quadratic equation, the solutions are given by:
[tex]S = \frac{-1cm \pm \sqrt{(1cm)^2 - 4*1*(-24cm)} }{2} \\\\S = \frac{-1cm \pm 12.04cm}{2}[/tex]
We only care for the positive solution, which is:
S = (-1cm + 12.04cm)/2 = 5.52cm
Concluding, the side length of the square is 5.52cm
If you want to learn more about systems of equations you can read:
https://brainly.com/question/13729904
