Sure, let's consider a real-world situation involving the area of a rectangular garden.
Imagine you have a rectangular garden whose length depends on its width. You are trying to find the possible widths of the garden that would result in a specific area, where the area [tex]\( A \)[/tex] of the garden is given by the function [tex]\( g(x) = 2x^2 - 13x + 6 \)[/tex].
To make this practical:
- [tex]\( x \)[/tex] represents the width of the garden in feet.
- [tex]\( g(x) \)[/tex] represents the area of the garden in square feet.
Suppose you are given that the area [tex]\( g(x) \)[/tex] is 6 square feet (as an example for simplicity). The viable equation then becomes:
[tex]\[ 2x^2 - 13x + 6 = 6 \][/tex]
This can be simplified, setting up the quadratic equation:
[tex]\[ 2x^2 - 13x + 6 - 6 = 0 \][/tex]
[tex]\[ 2x^2 - 13x = 0 \][/tex]
To find the widths [tex]\( x \)[/tex] that satisfy this scenario, we solve for [tex]\( x \)[/tex] using the quadratic formula where:
[tex]\[ a = 2, \, b = -13, \, \text{and} \, c = 6 \][/tex]
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Plugging in the coefficients, we solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4 \cdot 2 \cdot 6}}{2 \cdot 2} \][/tex]
After calculating this, we find the solutions:
[tex]\[ x_1 = 6.0 \, \text{feet}, \][/tex]
[tex]\[ x_2 = 0.5 \, \text{feet} \][/tex]
So, the possible widths of the garden that will make the area 6 square feet are either 6.0 feet or 0.5 feet. These widths are the dimensions that your garden can have to achieve the area of 6 square feet.
