Writing the Quadratic Equation

Oct 13, 2025 by leolavanderia.com.br 22 views

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Introduction

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

Let's break down the problem and understand what the Davidson family is trying to achieve. They want to extend the length and width of their patio by the same amount, which means they want to add a certain number of feet to both the length and the width. This will increase the total area of the patio by $160 \text{ ft}^2$ .

Setting Up the Equation

To solve this problem, we need to set up a quadratic equation. Let's assume that the amount by which the length and width are extended is $x$ feet. The new length of the patio will be $15 + x$ feet, and the new width will be $12 + x$ feet.

The area of the original patio is $15 \times 12 = 180 \text{ ft}^2$ . The area of the new patio will be $(15 + x)(12 + x)$ square feet. We want to find the value of $x$ that will increase the total area by $160 \text{ ft}^2$ .

Writing the Quadratic Equation

We can write the equation for the new area as:

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

Expanding the left-hand side of the equation, we get:

180 + 27x + x^2 = 340

Subtracting 180 from both sides of the equation, we get:

27x + x^2 = 160

Rearranging the equation to put it in standard quadratic form, we get:

x^2 + 27x - 160 = 0

Solving the Quadratic Equation

To solve this quadratic equation, we can use the quadratic formula:

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

In this case, $a = 1$ , $b = 27$ , and $c = -160$ . Plugging these values into the formula, we get:

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

Simplifying the expression under the square root, 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

We have two possible solutions for $x$ :

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

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

Interpreting the Solutions

The first solution, $x = 5$ , means that the Davidson family should extend the length and width of their patio by 5 feet each. This will increase the total area of the patio by $160 \text{ ft}^2$ .

The second solution, $x = -32$ , is not a valid solution in this context. Since the length and width of the patio cannot be negative, we can ignore this solution.

Conclusion

In conclusion, the Davidson family should extend the length and width of their patio by 5 feet each to increase the total area by $160 \text{ ft}^2$ . This problem can be solved using quadratic equations, which are a fundamental concept in mathematics.

Final Answer

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

Additional Information

The Davidson family's patio is currently 15 ft by 12 ft.
They want to extend the length and width by the same amount to increase the total area by $160 \text{ ft}^2$ .
The new length of the patio will be $15 + x$ feet, and the new width will be $12 + x$ feet.
The area of the original patio is $15 \times 12 = 180 \text{ ft}^2$ .
The area of the new patio will be $(15 + x)(12 + x)$ square feet.
The quadratic equation for the new area is $x^2 + 27x - 160 = 0$ .
The solutions to the quadratic equation are $x = 5$ and $x = -32$ .
The Davidson family should extend the length and width of their patio by 5 feet each to increase the total area by $160 \text{ ft}^2$ .

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Introduction

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.

Q: What is the current size of the Davidson family's patio?

A: The current size of the Davidson family's patio is 15 ft by 12 ft.

Q: What is the desired increase in the total area of the patio?

A: The Davidson family wants to increase the total area of the patio by $160 \text{ ft}^2$ .

Q: How much should the length and width of the patio be extended?

A: The length and width of the patio should be extended by 5 feet each.

Q: Why is the second solution, x = -32, not a valid solution?

A: The second solution, x = -32, is not a valid solution because the length and width of the patio cannot be negative.

Q: What is the new length and width of the patio after extension?

A: The new length of the patio will be $15 + 5 = 20$ feet, and the new width will be $12 + 5 = 17$ feet.

Q: What is the new area of the patio after extension?

A: The new area of the patio will be $(20)(17) = 340 \text{ ft}^2$ .

Q: How does the new area compare to the original area?

A: The new area is $340 \text{ ft}^2$ , which is an increase of $160 \text{ ft}^2$ from the original area of $180 \text{ ft}^2$ .

Q: What type of equation was used to solve this problem?

A: A quadratic equation was used to solve this problem.

Q: What is the formula for the quadratic equation?

A: The formula for the quadratic equation is $x^2 + bx + c = 0$ , where $a = 1$ , $b = 27$ , and $c = -160$ .

Q: How was the quadratic equation solved?

A: The quadratic equation was solved using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: What are the solutions to the quadratic equation?

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

Q: Why is the quadratic equation important in mathematics?

A: The quadratic equation is an important concept in mathematics because it can be used to solve a wide range of problems, including those involving quadratic relationships.

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

A: Quadratic equations have many real-world applications, including physics, engineering, economics, and computer science.

Q: How can quadratic equations be used in everyday life?

A: Quadratic equations can be used in everyday life to solve problems involving quadratic relationships, such as the Davidson family's patio expansion problem.

Conclusion

In conclusion, the Davidson family's patio expansion problem can be solved using quadratic equations. We hope this Q&A section has provided additional information and clarified any doubts. If you have any further questions, please don't hesitate to ask.

Final Answer

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

Additional Information

The Davidson family's patio is currently 15 ft by 12 ft.
They want to extend the length and width by the same amount to increase the total area by $160 \text{ ft}^2$ .
The new length of the patio will be $15 + x$ feet, and the new width will be $12 + x$ feet.
The area of the original patio is $15 \times 12 = 180 \text{ ft}^2$ .
The area of the new patio will be $(15 + x)(12 + x)$ square feet.
The quadratic equation for the new area is $x^2 + 27x - 160 = 0$ .
The solutions to the quadratic equation are $x = 5$ and $x = -32$ .
The Davidson family should extend the length and width of their patio by 5 feet each to increase the total area by $160 \text{ ft}^2$ .