The Davidson Family Wants To Expand Their Rectangular Patio, Which Currently Measures 15 Ft By 12 Ft. They Want To Extend The Length And Width By The Same Amount To Increase The Total Area Of The Patio By 160 Ft 2 160 \text{ Ft}^2 160 Ft 2 . Which Quadratic

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Introduction

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The Davidson family is looking to expand their rectangular patio, which currently measures 15 ft by 12 ft. They want to extend the length and width by the same amount to increase the total area of the patio by $160 \text{ ft}^2$. This problem can be solved using quadratic equations, which are a fundamental concept in mathematics.

Understanding the Problem

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Let's break down the problem and understand what the Davidson family is trying to achieve. They want to increase the total area of the patio by $160 \text{ ft}^2$, which means they want to add a certain amount of area to the existing patio. Since they want to extend the length and width by the same amount, we can assume that the new dimensions of the patio will be $(15 + x)$ ft by $(12 + x)$ ft, where $x$ is the amount by which they want to extend the length and width.

Setting Up the Equation

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The area of a rectangle is given by the formula $A = lw$, where $A$ is the area, $l$ is the length, and $w$ is the width. In this case, the area of the original patio is $15 \times 12 = 180 \text{ ft}^2$. The area of the new patio will be $(15 + x)(12 + x) \text{ ft}^2$. Since they want to increase the total area by $160 \text{ ft}^2$, we can set up the following equation:

$$(15 + x)(12 + x) - 180 = 160$$

Expanding and Simplifying the Equation

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To solve for $x$, we need to expand and simplify the equation. We can start by multiplying the two binomials:

$$(15 + x)(12 + x) = 180 + 15x + 12x + x^2$$

Simplifying the equation, we get:

$$180 + 27x + x^2 - 180 = 160$$

Combine like terms:

$$27x + x^2 - 160 = 0$$

Rearranging the Equation

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To make it easier to solve, we can rearrange the equation to get a quadratic equation in the form $ax^2 + bx + c = 0$. We can do this by subtracting $160$ from both sides of the equation:

$$x^2 + 27x - 160 = 0$$

Solving the Quadratic Equation

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Now that we have a quadratic equation, we can solve for $x$. There are several methods we can use to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, we can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Plugging in the values $a = 1$, $b = 27$, and $c = -160$, we get:

$$x = \frac{-27 \pm \sqrt{27^2 - 4(1)(-160)}}{2(1)}$$

Simplifying the equation, we get:

$$x = \frac{-27 \pm \sqrt{729 + 640}}{2}$$

$$x = \frac{-27 \pm \sqrt{1369}}{2}$$

$$x = \frac{-27 \pm 37}{2}$$

Finding the Solutions

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Now that we have the solutions to the quadratic equation, we can find the values of $x$. We have two possible solutions:

$$x = \frac{-27 + 37}{2} = \frac{10}{2} = 5$$

$$x = \frac{-27 - 37}{2} = \frac{-64}{2} = -32$$

Interpreting the Results

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The value of $x$ represents the amount by which the Davidson family wants to extend the length and width of their patio. Since $x$ cannot be negative, we can ignore the solution $x = -32$. Therefore, the Davidson family should extend the length and width of their patio by $5$ ft.

Conclusion

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In this problem, we used quadratic equations to solve for the amount by which the Davidson family wants to extend the length and width of their patio. We set up an equation based on the area of the original and new patios, expanded and simplified the equation, and solved for $x$ using the quadratic formula. The solution to the equation is $x = 5$, which represents the amount by which the Davidson family should extend the length and width of their patio.

Final Answer

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The final answer is $\boxed{5}$.

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Introduction

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In our previous article, we solved the Davidson family's patio expansion problem using quadratic equations. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the problem.

Q: What is the problem trying to solve?

A: The problem is trying to find the amount by which the Davidson family wants to extend the length and width of their patio to increase the total area by $160 \text{ ft}^2$.

Q: What are the original dimensions of the patio?

A: The original dimensions of the patio are $15$ ft by $12$ ft.

Q: What is the new area of the patio?

A: The new area of the patio is $(15 + x)(12 + x) \text{ ft}^2$, where $x$ is the amount by which the Davidson family wants to extend the length and width.

Q: How did you set up the equation?

A: We set up the equation by equating the area of the original patio to the area of the new patio and subtracting the original area from the new area to get the desired increase in area.

Q: How did you simplify the equation?

A: We simplified the equation by multiplying the two binomials and combining like terms.

Q: What method did you use to solve the quadratic equation?

A: We used the quadratic formula to solve the quadratic equation.

Q: What are the solutions to the quadratic equation?

A: The solutions to the quadratic equation are $x = 5$ and $x = -32$.

Q: Why did you ignore the solution $x = -32$?

A: We ignored the solution $x = -32$ because $x$ cannot be negative.

Q: What is the final answer?

A: The final answer is $x = 5$, which represents the amount by which the Davidson family should extend the length and width of their patio.

Q: What is the significance of the quadratic equation in this problem?

A: The quadratic equation is used to model the relationship between the original and new areas of the patio, and to find the amount by which the Davidson family wants to extend the length and width.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including modeling the trajectory of a projectile, finding the maximum or minimum value of a function, and solving problems involving area and volume.

Q: How can I apply quadratic equations to solve problems in my own life?

A: You can apply quadratic equations to solve problems in your own life by identifying the problem, setting up the equation, and using the quadratic formula to solve for the unknown variable.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include not simplifying the equation, not using the correct method to solve the equation, and not checking the solutions for validity.

Q: How can I practice solving quadratic equations?

A: You can practice solving quadratic equations by working through example problems, using online resources, and taking practice quizzes.

Q: What are some resources available to help me learn more about quadratic equations?

A: Some resources available to help you learn more about quadratic equations include textbooks, online tutorials, and video lectures.

Q: How can I apply quadratic equations to solve problems in mathematics and science?

A: You can apply quadratic equations to solve problems in mathematics and science by using the quadratic formula to solve for the unknown variable, and by using the equation to model real-world situations.

Q: What are some advanced topics related to quadratic equations?

A: Some advanced topics related to quadratic equations include quadratic inequalities, quadratic systems, and quadratic functions.

Q: How can I use quadratic equations to solve problems in engineering and physics?

A: You can use quadratic equations to solve problems in engineering and physics by using the quadratic formula to solve for the unknown variable, and by using the equation to model real-world situations.

Q: What are some common applications of quadratic equations in engineering and physics?

A: Some common applications of quadratic equations in engineering and physics include modeling the motion of objects, finding the maximum or minimum value of a function, and solving problems involving area and volume.

Q: How can I apply quadratic equations to solve problems in computer science?

A: You can apply quadratic equations to solve problems in computer science by using the quadratic formula to solve for the unknown variable, and by using the equation to model real-world situations.

Q: What are some common applications of quadratic equations in computer science?

A: Some common applications of quadratic equations in computer science include modeling the behavior of algorithms, finding the maximum or minimum value of a function, and solving problems involving area and volume.

Q: How can I use quadratic equations to solve problems in economics and finance?

A: You can use quadratic equations to solve problems in economics and finance by using the quadratic formula to solve for the unknown variable, and by using the equation to model real-world situations.

Q: What are some common applications of quadratic equations in economics and finance?

A: Some common applications of quadratic equations in economics and finance include modeling the behavior of economic systems, finding the maximum or minimum value of a function, and solving problems involving area and volume.

Q: How can I apply quadratic equations to solve problems in other fields?

A: You can apply quadratic equations to solve problems in other fields by using the quadratic formula to solve for the unknown variable, and by using the equation to model real-world situations.

Q: What are some common applications of quadratic equations in other fields?

A: Some common applications of quadratic equations in other fields include modeling the behavior of complex systems, finding the maximum or minimum value of a function, and solving problems involving area and volume.