91Ó°ÊÓ

Calculating the area of a rectangle is foundational in geometry. It's simple: multiply the length by the width. This formula can be written as:

• The Area, \( A \), of a rectangle = Length, \( l \), \( \times \) Width, \( w \)

In this particular problem, we're given the area of the garden as 200 square feet. With the length described in terms of the width, we have the equation:

• \( l \times w = 200 \)

This relationship helps us solve for unknown dimensions when either the length or width is expressed differently. In our case, the length was defined with respect to the width, forming a bridge to the next critical step.
Understanding how to set up equations from word problems is key. We translate descriptive text about dimensions into a useful mathematical equation, which then becomes solvable.

A quadratic equation appears when an area calculation transforms into multiplying a length and width with additional terms. Here, after substitution, the equation became \(3w^2 - w - 200 = 0\). Moving to a standard form helps us see if the equation can be factored more easily.
Factoring involves finding two binomials that multiply to give the original quadratic equation. The process is systematic:

• Multiply the leading coefficient by the constant term, which here yields \(-600\).
• Find two numbers that multiply to \(-600\) and sum to the middle term's coefficient, \(-1\).

These numbers, \(-25\) and \(24\), let us rewrite and group the equation for factoring:

• \(3w^2 + 24w - 25w - 200 = 0\).
• Grouping leads to \((3w + 24)(w - 25)\), where common factors let us simplify further.

By separating terms and spotting common factors, we can easily break down complicated equations into solvable parts.

Once a quadratic is factorable, solving it involves setting each factor to zero. For \((3w - 25)(w + 8) = 0\), set each binomial equal to zero to find potential solutions:

• \(3w - 25 = 0\) implies \(w = \frac{25}{3}\)
• \(w + 8 = 0\) implies \(w = -8\), which isn't feasible in geometry due to negative dimensions.

Since we discard \(-8\), \(w = \frac{25}{3}\), approximately 8.33 feet. Then, substitute \(w\) back into the expression for length, \(l = 3w - 1\):

• \(l = 3 \times \frac{25}{3} - 1 = 24\) feet

This final calculation gives the length of the garden. Solving factorized equations requires verifying each solution and considering real-world constraints like non-negative measurements.