Tople "6i Applications \( \begin{array}{l}\text { 50. The dimensions of a rectangle can be represented by the expressions }(x+7) \text { and }(x-3) \text {. If the } \\ \text { of the rectangle is } 75 \text { square feet, find the value of } x \text {. }\end{array} \)

To find the value of \( x \), we start by setting up the equation for the area of the rectangle, which is given by multiplying the two dimensions: \[ (x + 7)(x - 3) = 75. \] Expanding the left side, we have: \[ x^2 - 3x + 7x - 21 = 75, \] which simplifies to: \[ x^2 + 4x - 21 = 75. \] Next, we move 75 to the left side of the equation to set it to zero: \[ x^2 + 4x - 96 = 0. \] Applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 4, c = -96 \): \[ b^2 - 4ac = 4^2 - 4(1)(-96) = 16 + 384 = 400. \] Now calculate \( x \): \[ x = \frac{-4 \pm \sqrt{400}}{2(1)} = \frac{-4 \pm 20}{2}. \] This gives us two possible solutions: 1. \( x = \frac{16}{2} = 8 \) 2. \( x = \frac{-24}{2} = -12 \) (not applicable, as dimensions can't be negative) Thus, the value of \( x \) is 8. Now, let’s cover some interesting facts! Did you know that the method of solving quadratic equations, like we just did, was significantly developed during the Islamic Golden Age? Mathematicians like Al-Khwarizmi laid the groundwork for algebraic techniques we use today, showcasing how this ancient knowledge continues to impact modern math education. In practical scenarios, understanding how to manipulate equations and find dimensions can drastically help in fields like architecture and engineering. For instance, if you're designing a park or building, knowing how to calculate areas efficiently ensures that you make optimal use of space while staying within your material budget. It's about turning numbers into real-world solutions!