- Expand the equation \((6+x)(9+x)=88\) to standard form: \(x^2 + 15x - 34 = 0\).
- Solve the equation by factoring to find the roots: \((x + 17)(x - 2) = 0\), so \(x = 2\) or \(x = -17\).
- Recognize that \(x\) represents the uniform increase added to both dimensions.
- Conclude that the values represent the two possible mathematical amounts by which the dimensions could be changed, thus the correct option is (a). \(\boxed{a}\)
Explanation
- Analyzing the Problem and the Equation
The problem asks us to consider a rectangular patio that is initially \(9\text{ ft}\) by \(6\text{ ft}\). We are told that both the length and the width are increased by the same amount, \(x\), such that the new area is \(88\text{ sq ft}\). This is described by the equation \((6+x)(9+x)=88\). To understand what \(x\) represents, we first expand and solve this quadratic equation.
- Solving the Quadratic Equation
Expanding the left side of the equation \((6+x)(9+x)=88\):
\[x^2 + 9x + 6x + 54 = 88\]
\[x^2 + 15x + 54 = 88\]
Subtracting 88 from both sides gives:
\[x^2 + 15x - 34 = 0\]
We can solve this using the zero product property by factoring the quadratic expression:
\[(x + 17)(x - 2) = 0\]
The two roots of this equation are \(x = -17\) and \(x = 2\). These values represent the mathematical solutions for the change in dimensions required to reach the area of 88 square feet.
- Interpreting the Result
In the context of the physical dimensions of the patio, \(x\) represents the amount of increase applied to both the length and the width. Mathematically, the solutions \(x = 2\) and \(x = -17\) are the values that satisfy the equation. While only \(x = 2\) provides a physically meaningful positive increase, the variable \(x\) itself represents the amount of change added to the dimensions. Therefore, the solutions for \(x\) represent two potential (mathematical) amounts by which the dimensions were changed. Comparing this to the given options, option (a) "two possible amounts by which the dimensions were changed" is the correct interpretation.
Examples
Suppose you are planning to extend a small garden plot to grow more vegetables. If you increase the length and width of your rectangular garden by the same amount to reach a specific total area, you can use a quadratic equation to find exactly how many feet to add to each side. This helps in efficient land use and resource management, ensuring your garden expansion fits exactly within your available backyard space.