- Identify the area as the product of two linear factors: \(A = (x+m)(x+n) = x^2 + (m+n)x + mn\).
- Match the quadratic expression \(x^2 - 4x - 12\) by finding \(m, n\) such that \(m+n = -4\) and \(mn = -12\).
- Determine that the factors are \(m=2\) and \(n=-6\).
- Conclude that the side lengths are \((x+2)\) and \((x-6)\), so the answer is \(\boxed{(x+2) \text{ and } (x-6)}\).
Explanation
- Analyzing the Problem
The problem asks for the side lengths of a rectangle whose area is given by the quadratic expression \(A = x^2 - 4x - 12\). Since the area of a rectangle is the product of its length and its width, the side lengths are the factors of this expression. We need to express \(x^2 - 4x - 12\) in the form \((x + m)(x + n)\), where the product of the constants \(m\) and \(n\) is \(-12\) and their sum is \(-4\).
- Factoring the Expression
To find the values of \(m\) and \(n\), we look for two numbers that multiply to \(-12\) and add up to \(-4\). Listing pairs of factors of \(-12\):
- \(1\) and \(-12\): sum = \(-11\)
- \(-1\) and \(12\): sum = \(11\)
- \(2\) and \(-6\): sum = \(-4\)
- \(-2\) and \(6\): sum = \(4\)
- \(3\) and \(-4\): sum = \(-1\)
- \(-3\) and \(4\): sum = \(1\)
The pair that sums to \(-4\) is \(2\) and \(-6\). Therefore, the quadratic expression can be factored as \((x + 2)(x - 6)\).
- Verification and Conclusion
We verify our result by expanding the product \((x+2)(x-6)\):
\((x+2)(x-6) = x \cdot x + x \cdot (-6) + 2 \cdot x + 2 \cdot (-6) = x^2 - 6x + 2x - 12 = x^2 - 4x - 12\).
This expansion matches our original expression. Comparing this to the given options, we identify the side lengths as \((x+2)\) and \((x-6)\).
Examples
In geometry, finding the side lengths of a rectangular area is a practical application of factoring quadratic equations. For example, if you are designing a garden plot or a room floor plan where the total space is defined by an algebraic model, factoring allows you to determine the specific lengths and widths needed for construction. By mastering this skill, you can easily translate complex area requirements into simple, actionable dimensions for any rectangular space.