Final answer:
To solve the rectangular picture problem, we set x as the length, resulting in the equation x(x - 4) = 96. Factoring the quadratic equation yields x = 12 inches for the length, and thus the width is x - 4 = 8 inches, solving the mathematical problem completely.
Step-by-step explanation:
The question asks us to solve the mathematical problem of finding the dimensions of a rectangular picture where the width is 4 inches less than the length and the area is 96 square inches. Although the provided information gives an example with squares, we can apply similar reasoning to solve the rectangle problem.
Let x represent the length of the picture. Then, the width would be x - 4 inches. The area of a rectangle is found by multiplying the length by the width, which gives us the equation:
x(x - 4) = 96
Expanding this equation:
x^2 - 4x = 96
To solve this quadratic equation, we'll move all terms to one side:
x^2 - 4x - 96 = 0
Next, we factor this quadratic equation:
(x - 12)(x + 8) = 0
This gives us two possible solutions for x, namely 12 and -8. However, since a negative length doesn't make sense in this context, we will use the positive value:
x = 12 inches (length)
Now we substitute x back into the equation to find the width:
Width = x - 4 = 12 - 4 = 8 inches
Therefore, the dimensions of the picture are 12 inches by 8 inches.