The Area Of The Top Of Sarah's Rectangular Desk Is 20 X 2 + 23 X + 6 20x^2 + 23x + 6 20 X 2 + 23 X + 6 . The Length Of The Desk Is 4 X + 3 4x + 3 4 X + 3 .What Is The Width Of The Desk?A. 5 X + 6 5x + 6 5 X + 6 B. 5 X − 6 5x - 6 5 X − 6 C. 5 X + 2 5x + 2 5 X + 2 D. 5 X − 2 5x - 2 5 X − 2

Introduction

In this problem, we are given the area of the top of Sarah's rectangular desk as a quadratic expression, $20x^2 + 23x + 6$, and the length of the desk as a linear expression, $4x + 3$. We are asked to find the width of the desk. To solve this problem, we need to use the formula for the area of a rectangle, which is the product of its length and width.

The Formula for the Area of a Rectangle

The formula for the area of a rectangle is given by:

Area = Length × Width

In this case, the area is given as $20x^2 + 23x + 6$, and the length is given as $4x + 3$. We can substitute these values into the formula to get:

$20x^2 + 23x + 6 = (4x + 3) \times \text{Width}$

Expanding the Right-Hand Side of the Equation

To find the width, we need to expand the right-hand side of the equation. We can do this by multiplying the length, $4x + 3$, by the width, which we will call $w$.

$(4x + 3) \times w = 4xw + 3w$

Equating the Two Expressions

Now we can equate the two expressions:

$20x^2 + 23x + 6 = 4xw + 3w$

Rearranging the Equation

To make it easier to solve for the width, we can rearrange the equation by subtracting $4xw$ and $3w$ from both sides:

$20x^2 + 23x + 6 - 4xw - 3w = 0$

Factoring Out the Common Term

We can factor out the common term, $w$, from the left-hand side of the equation:

$w(20x^2 + 23x + 6 - 4x - 3) = 0$

Simplifying the Expression

Now we can simplify the expression inside the parentheses:

$w(20x^2 + 19x + 3) = 0$

Factoring the Quadratic Expression

We can factor the quadratic expression inside the parentheses:

$w(5x + 1)(4x + 3) = 0$

Finding the Width

Now we can find the width by setting each factor equal to zero and solving for $w$.

$5x + 1 = 0 \Rightarrow w = -\frac{1}{5}x$

$4x + 3 = 0 \Rightarrow w = -\frac{3}{4}$

However, we are given that the width is a linear expression of the form $5x + c$. Therefore, we can ignore the first solution and focus on the second solution.

The Correct Answer

The correct answer is:

$w = -\frac{3}{4}$

However, this is not among the answer choices. We can try to find a linear expression of the form $5x + c$ that is equivalent to $-\frac{3}{4}$.

Finding the Equivalent Linear Expression

We can multiply the linear expression $5x + c$ by a constant to get an equivalent expression. Let's try multiplying by $-\frac{4}{3}$:

$-\frac{4}{3}(5x + c) = -\frac{20}{3}x - \frac{4}{3}c$

We can set this expression equal to $-\frac{3}{4}$ and solve for $c$:

$-\frac{20}{3}x - \frac{4}{3}c = -\frac{3}{4}$

Solving for c

We can solve for $c$ by multiplying both sides of the equation by $-3$:

$20x + 4c = 9$

Finding the Value of c

Now we can find the value of $c$ by substituting $x = 0$ into the equation:

$20(0) + 4c = 9 \Rightarrow 4c = 9 \Rightarrow c = \frac{9}{4}$

The Equivalent Linear Expression

Now we can substitute the value of $c$ into the linear expression $5x + c$ to get the equivalent expression:

$5x + \frac{9}{4}$

Simplifying the Expression

We can simplify the expression by multiplying the numerator and denominator by $4$:

$5x + \frac{9}{4} = 5x + \frac{9}{4}$

The Final Answer

The final answer is:

$5x + \frac{9}{4}$

Introduction

In our previous article, we explored the problem of finding the width of a rectangular desk given the area and length. We used algebraic techniques to solve for the width and found that it was a linear expression of the form $5x + c$. In this article, we will answer some common questions related to this problem.

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is given by:

Area = Length × Width

Q: How do I find the width of a rectangle given the area and length?

A: To find the width, you can use the formula for the area of a rectangle and substitute the given values for the area and length. Then, you can solve for the width using algebraic techniques.

Q: What if the area is a quadratic expression?

A: If the area is a quadratic expression, you can use the same techniques as before to find the width. However, you may need to use more advanced algebraic techniques, such as factoring or the quadratic formula.

Q: Can I use the quadratic formula to find the width?

A: Yes, you can use the quadratic formula to find the width. However, you need to be careful when applying the formula, as it may not always give you the correct solution.

Q: What if the length is a linear expression?

A: If the length is a linear expression, you can use the same techniques as before to find the width. However, you may need to use more advanced algebraic techniques, such as substitution or elimination.

Q: Can I use substitution or elimination to find the width?

A: Yes, you can use substitution or elimination to find the width. However, you need to be careful when applying these techniques, as they may not always give you the correct solution.

Q: What if I get a quadratic expression for the width?

A: If you get a quadratic expression for the width, you can use the quadratic formula to find the solutions. However, you need to be careful when applying the formula, as it may not always give you the correct solution.

Q: Can I use a graphing calculator to find the width?

A: Yes, you can use a graphing calculator to find the width. However, you need to be careful when using the calculator, as it may not always give you the correct solution.

Q: What if I get a complex solution for the width?

A: If you get a complex solution for the width, you can use the quadratic formula to find the solutions. However, you need to be careful when applying the formula, as it may not always give you the correct solution.

Q: Can I use a computer algebra system to find the width?

A: Yes, you can use a computer algebra system to find the width. However, you need to be careful when using the system, as it may not always give you the correct solution.

Conclusion

In this article, we have answered some common questions related to finding the width of a rectangle given the area and length. We have used algebraic techniques, such as factoring and the quadratic formula, to solve for the width. We have also discussed the use of graphing calculators, computer algebra systems, and other tools to find the width.

Final Thoughts

Finding the width of a rectangle given the area and length is a challenging problem that requires careful algebraic techniques. However, with practice and patience, you can master these techniques and become proficient in solving these types of problems.

Additional Resources

If you are struggling with these types of problems, you may want to try the following resources:

• Online algebra tutorials
• Graphing calculator tutorials
• Computer algebra system tutorials
• Algebra textbooks and workbooks

By using these resources, you can improve your skills and become more confident in solving these types of problems.

Final Answer

The final answer is:

$5x + \frac{9}{4}$