- Multiply the two linear binomials \(g(x)\) and \(h(x)\) using the distributive property.
- Calculate each of the four components of the expansion: \(\frac{18}{5}x^2\), \(-3x\), \(7x\), and \(-\frac{35}{6}\).
- Combine the linear terms \(-3x\) and \(7x\) to get \(4x\).
- The final equivalent quadratic expression is \(\boxed{\frac{18}{5}x^2 + 4x - \frac{35}{6}}\).
Explanation
- Understanding the Problem
We are given two functions defined by the equations \(g(x) = \frac{3}{5}x + \frac{7}{6}\) and \(h(x) = 6x - 5\). The objective is to find the expression equivalent to the product \(g(x) \cdot h(x)\), which is:
\[g(x) \cdot h(x) = \left(\frac{3}{5}x + \frac{7}{6}\right)(6x - 5)\]
This involves multiplying two linear binomials, resulting in a quadratic expression of the form \(Ax^2 + Bx + C\).
- Applying the Distributive Property
To calculate the product, we use the distributive property (often called the FOIL method, which stands for First, Outer, Inner, Last):
\[g(x) \cdot h(x) = \left(\frac{3}{5}x \cdot 6x\right) + \left(\frac{3}{5}x \cdot -5\right) + \left(\frac{7}{6} \cdot 6x\right) + \left(\frac{7}{6} \cdot -5\right)\]
We calculate each of these four products individually:
- First terms: \(\frac{3}{5}x \cdot 6x = \frac{18}{5}x^2\)
- Outer terms: \(\frac{3}{5}x \cdot -5 = -3x\)
- Inner terms: \(\frac{7}{6} \cdot 6x = 7x\)
- Last terms: \(\frac{7}{6} \cdot -5 = -\frac{35}{6}\)
- Combining Like Terms and Final Result
Now we substitute these results back into the expression and combine like terms (the terms containing \(x\)):
\[g(x) \cdot h(x) = \frac{18}{5}x^2 - 3x + 7x - \frac{35}{6}\]
Combining the terms \(-3x\) and \(7x\), we get \(4x\). Thus, the final quadratic expression is:
\[g(x) \cdot h(x) = \frac{18}{5}x^2 + 4x - \frac{35}{6}\]
Examples
In real-world applications like economics or engineering, multiplying functions is a common task. For example, if a company's price per unit \(p(x)\) and demand \(d(x)\) are both expressed as linear functions of the number of units sold \(x\), the revenue \(R(x) = p(x) \cdot d(x)\) becomes a quadratic function. Determining the correct expression for revenue allows businesses to analyze their profit potential and identify the optimal production levels. Here, we used algebraic multiplication to find a model that represents the combined growth of two linear factors.