Jillian Is Trying To Save Water, So She Reduces The Size Of Her Square Grass Lawn By 8 Feet On Each Side. The Area Of The Smaller Lawn Is 144 Square Feet. In The Equation \[$(x-8)^2=144\$\], \[$x\$\] Represents The Side Measure Of The
Introduction
As the world grapples with the challenges of water conservation, individuals are taking steps to reduce their water usage. One way to achieve this is by reducing the size of lawns, which are often a significant consumer of water. In this article, we will explore a real-life scenario where Jillian reduces the size of her square grass lawn by 8 feet on each side, resulting in a smaller lawn with an area of 144 square feet. We will use this scenario to solve a mathematical equation and find the side measure of the smaller lawn.
The Problem
Jillian's original lawn was a square with an unknown side measure. After reducing the size of the lawn by 8 feet on each side, the new lawn has an area of 144 square feet. We can represent the side measure of the smaller lawn as x. The equation {(x-8)^2=144$}$ represents the relationship between the side measure of the smaller lawn and its area.
Understanding the Equation
The equation {(x-8)^2=144$}$ is a quadratic equation, which is a polynomial equation of degree two. The general form of a quadratic equation is {ax^2+bx+c=0$}$, where a, b, and c are constants. In this case, the equation is {(x-8)^2=144$}$, which can be expanded to {x^2-16x+64=144$}$.
Solving the Equation
To solve the equation {x^2-16x+64=144$}$, we can start by subtracting 144 from both sides of the equation, resulting in {x^2-16x-80=0$}$. This is a quadratic equation in the form of {ax^2+bx+c=0$}$, where a = 1, b = -16, and c = -80.
Using the Quadratic Formula
The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. The formula is {x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$}$. In this case, a = 1, b = -16, and c = -80. Plugging these values into the formula, we get {x=\frac{-(-16)\pm\sqrt{(-16)^2-4(1)(-80)}}{2(1)}$}$.
Simplifying the Equation
Simplifying the equation, we get {x=\frac{16\pm\sqrt{256+320}}{2}$}$. This can be further simplified to {x=\frac{16\pm\sqrt{576}}{2}$}$.
Finding the Solutions
The square root of 576 is 24. Plugging this value back into the equation, we get {x=\frac16\pm24}{2}$}$. This results in two possible solutions{2}=20$}$ and {x=\frac{16-24}{2}=-4$}$.
Interpreting the Results
The solution {x=20$}$ represents the side measure of the smaller lawn. This means that the side measure of the original lawn was 20 + 8 = 28 feet. The solution {x=-4$}$ is not a valid solution in this context, as the side measure of a lawn cannot be negative.
Conclusion
In this article, we used a real-life scenario to solve a mathematical equation and find the side measure of a smaller lawn. We started with the equation {(x-8)^2=144$}$ and used the quadratic formula to find the solutions. The solution {x=20$}$ represents the side measure of the smaller lawn, while the solution {x=-4$}$ is not a valid solution in this context. This problem demonstrates the importance of understanding and applying mathematical concepts to real-world problems.
Additional Resources
For more information on quadratic equations and the quadratic formula, please refer to the following resources:
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Formula
- Wolfram Alpha: Quadratic Equation Solver
Final Thoughts
Q: What is the main goal of Jillian's lawn reduction?
A: The main goal of Jillian's lawn reduction is to save water by reducing the size of her square grass lawn.
Q: What is the original area of Jillian's lawn?
A: The original area of Jillian's lawn is not explicitly stated in the problem, but we can find it by using the equation {(x-8)^2=144$}$. We know that the area of the smaller lawn is 144 square feet, and we can use this information to find the original area.
Q: How do we find the original area of Jillian's lawn?
A: To find the original area of Jillian's lawn, we can use the equation {(x-8)^2=144$}$. We know that the area of the smaller lawn is 144 square feet, and we can use this information to find the original area. We can start by adding 64 to both sides of the equation, resulting in {x^2-16x+64+64=144+64$}$. This simplifies to {x^2-16x+128=208$}$.
Q: How do we solve for x in the equation {x^2-16x+128=208$}$?
A: To solve for x in the equation {x^2-16x+128=208$}$, we can start by subtracting 208 from both sides of the equation, resulting in {x^2-16x-80=0$}$. This is a quadratic equation in the form of {ax^2+bx+c=0$}$, where a = 1, b = -16, and c = -80.
Q: How do we find the solutions to the quadratic equation {x^2-16x-80=0$}$?
A: To find the solutions to the quadratic equation {x^2-16x-80=0$}$, we can use the quadratic formula: {x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$}$. In this case, a = 1, b = -16, and c = -80. Plugging these values into the formula, we get {x=\frac{-(-16)\pm\sqrt{(-16)^2-4(1)(-80)}}{2(1)}$}$.
Q: What are the solutions to the quadratic equation {x^2-16x-80=0$}$?
A: The solutions to the quadratic equation {x^2-16x-80=0$}$ are {x=\frac16\pm\sqrt{576}}{2}$}$. This simplifies to {x=\frac{16\pm24}{2}$}$. This results in two possible solutions{2}=20$}$ and {x=\frac{16-24}{2}=-4$}$.
Q: What does the solution {x=20$}$ represent?
A: The solution {x=20$}$ represents the side measure of the original lawn.
Q: What does the solution {x=-4$}$ represent?
A: The solution {x=-4$}$ is not a valid solution in this context, as the side measure of a lawn cannot be negative.
Q: How do we find the original area of Jillian's lawn?
A: To find the original area of Jillian's lawn, we can use the equation {(x-8)^2=144$}$. We know that the area of the smaller lawn is 144 square feet, and we can use this information to find the original area. We can start by adding 64 to both sides of the equation, resulting in {x^2-16x+64+64=144+64$}$. This simplifies to {x^2-16x+128=208$}$.
Q: How do we solve for x in the equation {x^2-16x+128=208$}$?
A: To solve for x in the equation {x^2-16x+128=208$}$, we can start by subtracting 208 from both sides of the equation, resulting in {x^2-16x-80=0$}$. This is a quadratic equation in the form of {ax^2+bx+c=0$}$, where a = 1, b = -16, and c = -80.
Q: What is the original area of Jillian's lawn?
A: The original area of Jillian's lawn is 28^2 = 784 square feet.
Q: What is the final answer to the problem?
A: The final answer to the problem is that the side measure of the original lawn is 28 feet, and the original area of Jillian's lawn is 784 square feet.