A Rectangular Kitchen Rug Has A Length Of $(2x^2 - 6x + 7$\] And A Width Of $(x + 3$\]. What Is The Area Of The Rug In Simplest Form?
**A rectangular kitchen rug has a length of $(2x^2 - 6x + 7)$ and a width of $(x + 3)$. What is the area of the rug in simplest form?**
Understanding the Problem
To find the area of the rectangular kitchen rug, we need to multiply its length and width. The length is given by the quadratic expression , and the width is given by the linear expression .
Step 1: Multiply the Length and Width
To find the area, we multiply the length and width of the rug. We can do this by multiplying the two expressions together:
Area = Length × Width =
Step 2: Expand the Expression
To simplify the expression, we need to expand it by multiplying each term in the first expression by each term in the second expression:
= =
Step 3: Simplify the Expression
Now, we can simplify the expression by combining like terms:
= =
The Final Answer
Therefore, the area of the rectangular kitchen rug in simplest form is:
Frequently Asked Questions
Q: What is the formula for finding the area of a rectangle?
A: The formula for finding the area of a rectangle is Area = Length × Width.
Q: How do I multiply a quadratic expression by a linear expression?
A: To multiply a quadratic expression by a linear expression, you need to multiply each term in the quadratic expression by each term in the linear expression.
Q: What is the difference between expanding and simplifying an expression?
A: Expanding an expression involves multiplying each term in one expression by each term in another expression, while simplifying an expression involves combining like terms to reduce the expression to its simplest form.
Q: Can I use a calculator to find the area of the rug?
A: Yes, you can use a calculator to find the area of the rug. However, it's always a good idea to simplify the expression first to make sure you get the correct answer.
Q: What if the expressions are more complex?
A: If the expressions are more complex, you may need to use algebraic techniques such as factoring or using the distributive property to simplify the expression before multiplying it.
Q: Can I use this method to find the area of any shape?
A: No, this method is specifically for finding the area of a rectangle. If you need to find the area of a different shape, you will need to use a different formula or method.
Q: What if I make a mistake in my calculations?
A: If you make a mistake in your calculations, you may end up with an incorrect answer. To avoid this, make sure to double-check your work and use algebraic techniques such as factoring or using the distributive property to simplify the expression before multiplying it.
Q: Can I use this method to find the perimeter of the rug?
A: No, this method is specifically for finding the area of a rectangle. If you need to find the perimeter of the rug, you will need to use a different formula or method.
Q: What if I need to find the area of a rug with a non-linear width?
A: If the width of the rug is not linear, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a non-rectangular shape?
A: No, this method is specifically for finding the area of a rectangle. If you need to find the area of a rug with a non-rectangular shape, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a variable width?
A: If the width of the rug is variable, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a negative width?
A: No, this method is specifically for finding the area of a rectangle with a positive width. If the width of the rug is negative, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a complex width?
A: If the width of the rug is complex, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a fractional width?
A: No, this method is specifically for finding the area of a rectangle with a whole number width. If the width of the rug is fractional, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a width that is a function of x?
A: If the width of the rug is a function of x, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a width that is a function of y?
A: No, this method is specifically for finding the area of a rectangle with a width that is a function of x. If the width of the rug is a function of y, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a width that is a function of both x and y?
A: If the width of the rug is a function of both x and y, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a non-linear length?
A: No, this method is specifically for finding the area of a rectangle with a linear length. If the length of the rug is non-linear, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a length that is a function of x?
A: If the length of the rug is a function of x, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a length that is a function of y?
A: No, this method is specifically for finding the area of a rectangle with a length that is a function of x. If the length of the rug is a function of y, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a length that is a function of both x and y?
A: If the length of the rug is a function of both x and y, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a non-rectangular shape?
A: No, this method is specifically for finding the area of a rectangle. If you need to find the area of a rug with a non-rectangular shape, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a variable shape?
A: If the shape of the rug is variable, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a negative shape?
A: No, this method is specifically for finding the area of a rectangle with a positive shape. If the shape of the rug is negative, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a complex shape?
A: If the shape of the rug is complex, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.
Q: Can I use this method to find the area of a rug with a fractional shape?
A: No, this method is specifically for finding the area of a rectangle with a whole number shape. If the shape of the rug is fractional, you will need to use a different method or formula.
Q: What if I need to find the area of a rug with a shape that is a function of x?
A: If the shape of the rug is a function of x, you will need to use a different method to find the area. This may involve using calculus or other advanced mathematical techniques.