12. A Wall Contains A Rectangular Window. The Area (in Square Feet) Of The Window Is Represented By $x^2 - 8x + 15$.a. Write A Binomial That Represents The Height Of The Window.Answer: $(x - 5)$b. Find The Perimeter Of The Window

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Introduction

In this problem, we are given a quadratic expression representing the area of a rectangular window in square feet. The area is given by the expression $x^2 - 8x + 15$. Our goal is to find the binomial that represents the height of the window and then determine the perimeter of the window.

Understanding the Area of the Window

The area of the window is represented by the quadratic expression $x^2 - 8x + 15$. To find the binomial that represents the height of the window, we need to factor the given quadratic expression.

Factoring the Quadratic Expression

To factor the quadratic expression $x^2 - 8x + 15$, we need to find two numbers whose product is 15 and whose sum is -8. These numbers are -5 and -3, since (-5) × (-3) = 15 and (-5) + (-3) = -8.

Using this information, we can rewrite the quadratic expression as:

x2−8x+15=(x−5)(x−3)x^2 - 8x + 15 = (x - 5)(x - 3)

Finding the Binomial that Represents the Height of the Window

From the factored form of the quadratic expression, we can see that the binomial that represents the height of the window is $(x - 5)$.

Understanding the Relationship Between the Area and the Dimensions of the Window

The area of a rectangle is given by the product of its length and width. In this case, the area is represented by the quadratic expression $x^2 - 8x + 15$, and the binomial $(x - 5)$ represents the height of the window.

Finding the Perimeter of the Window

To find the perimeter of the window, we need to find the length and width of the window. Since the area is given by the product of the length and width, we can set up the following equation:

Area = Length × Width

Substituting the given area and the binomial that represents the height of the window, we get:

x2−8x+15=x×(x−5)x^2 - 8x + 15 = x \times (x - 5)

Simplifying the equation, we get:

x2−8x+15=x2−5xx^2 - 8x + 15 = x^2 - 5x

Subtracting $x^2$ from both sides of the equation, we get:

−8x+15=−5x-8x + 15 = -5x

Adding 5x to both sides of the equation, we get:

−3x+15=0-3x + 15 = 0

Subtracting 15 from both sides of the equation, we get:

−3x=−15-3x = -15

Dividing both sides of the equation by -3, we get:

x=5x = 5

Finding the Length of the Window

Now that we have found the value of x, we can find the length of the window by substituting x into the binomial that represents the height of the window:

Length = x - 5 = 5 - 5 = 0

However, this is not possible, as the length of the window cannot be zero. This means that our previous assumption that the binomial $(x - 5)$ represents the height of the window is incorrect.

Revisiting the Factored Form of the Quadratic Expression

Let's revisit the factored form of the quadratic expression:

x2−8x+15=(x−5)(x−3)x^2 - 8x + 15 = (x - 5)(x - 3)

We can see that the binomial $(x - 5)$ actually represents the width of the window, not the height.

Finding the Height of the Window

Now that we have found the width of the window, we can find the height of the window by substituting x into the binomial that represents the width of the window:

Width = x - 5 = 5 - 5 = 0

However, this is not possible, as the width of the window cannot be zero. This means that our previous assumption that the binomial $(x - 5)$ represents the width of the window is incorrect.

Revisiting the Factored Form of the Quadratic Expression Again

Let's revisit the factored form of the quadratic expression again:

x2−8x+15=(x−5)(x−3)x^2 - 8x + 15 = (x - 5)(x - 3)

We can see that the binomial $(x - 3)$ actually represents the width of the window, not the height.

Finding the Height of the Window Again

Now that we have found the width of the window, we can find the height of the window by substituting x into the binomial that represents the width of the window:

Width = x - 3 = 5 - 3 = 2

Finding the Perimeter of the Window

Now that we have found the length and width of the window, we can find the perimeter of the window by using the formula:

Perimeter = 2 × (Length + Width)

Substituting the values of length and width, we get:

Perimeter = 2 × (5 + 2) = 2 × 7 = 14

Conclusion

In this problem, we were given a quadratic expression representing the area of a rectangular window in square feet. We found the binomial that represents the height of the window and then determined the perimeter of the window. We learned that the binomial $(x - 5)$ actually represents the width of the window, not the height, and that the binomial $(x - 3)$ actually represents the height of the window, not the width. We also learned that the perimeter of the window is 14 feet.

Introduction

In our previous article, we explored the problem of a wall containing a rectangular window with an area represented by the quadratic expression $x^2 - 8x + 15$. We found the binomial that represents the height of the window and then determined the perimeter of the window. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the binomial that represents the height of the window?

A: The binomial that represents the height of the window is actually $(x - 3)$, not $(x - 5)$.

Q: How do I find the perimeter of the window?

A: To find the perimeter of the window, you need to find the length and width of the window. Once you have the length and width, you can use the formula: Perimeter = 2 × (Length + Width).

Q: What is the length of the window?

A: The length of the window is 5 feet.

Q: What is the width of the window?

A: The width of the window is 2 feet.

Q: How do I find the area of the window?

A: To find the area of the window, you can use the formula: Area = Length × Width. In this case, the area is given by the quadratic expression $x^2 - 8x + 15$.

Q: What is the relationship between the area and the dimensions of the window?

A: The area of a rectangle is given by the product of its length and width. In this case, the area is represented by the quadratic expression $x^2 - 8x + 15$, and the binomial $(x - 3)$ represents the height of the window.

Q: Can I use the binomial $(x - 5)$ to find the height of the window?

A: No, you cannot use the binomial $(x - 5)$ to find the height of the window. The binomial $(x - 5)$ actually represents the width of the window, not the height.

Q: Can I use the binomial $(x - 3)$ to find the width of the window?

A: No, you cannot use the binomial $(x - 3)$ to find the width of the window. The binomial $(x - 3)$ actually represents the height of the window, not the width.

Q: How do I know which binomial represents the height of the window and which binomial represents the width of the window?

A: To determine which binomial represents the height of the window and which binomial represents the width of the window, you need to look at the factored form of the quadratic expression. In this case, the factored form of the quadratic expression is $(x - 5)(x - 3)$, which means that the binomial $(x - 5)$ represents the width of the window and the binomial $(x - 3)$ represents the height of the window.

Conclusion

In this article, we answered some frequently asked questions related to the problem of a wall containing a rectangular window with an area represented by the quadratic expression $x^2 - 8x + 15$. We learned that the binomial $(x - 5)$ actually represents the width of the window, not the height, and that the binomial $(x - 3)$ actually represents the height of the window, not the width. We also learned how to find the perimeter of the window and how to determine which binomial represents the height of the window and which binomial represents the width of the window.