Candy Draws A Square Design With A Side Length Of $x$ Inches For The Window At The Pet Shop. She Takes The Design To The Printer And Asks For A Sign That Has An Area Of $16x^2 - 40x + 25$ Square Inches.What Is The Side Length, In

Introduction

When it comes to designing a sign for a pet shop, the area of the sign is crucial in determining its size. In this scenario, Candy wants a sign with an area of $16x^2 - 40x + 25$ square inches. To find the side length of the square sign, we need to use the formula for the area of a square, which is $A = s^2$, where $A$ is the area and $s$ is the side length. We will use algebraic techniques to solve for the side length.

Understanding the Problem

The problem states that the area of the sign is given by the quadratic expression $16x^2 - 40x + 25$. To find the side length, we need to find the value of $x$ that satisfies the equation $16x^2 - 40x + 25 = s^2$. Since the area of a square is equal to the square of its side length, we can set up the equation $16x^2 - 40x + 25 = s^2$.

Solving the Quadratic Equation

To solve the quadratic equation $16x^2 - 40x + 25 = s^2$, we can start by rearranging the equation to get $16x^2 - 40x + (25 - s^2) = 0$. This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 16$, $b = -40$, and $c = 25 - s^2$. We can use the quadratic formula to solve for $x$.

The Quadratic Formula

The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 16$, $b = -40$, and $c = 25 - s^2$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-40) \pm \sqrt{(-40)^2 - 4(16)(25 - s^2)}}{2(16)}$

Simplifying the expression, we get:

$x = \frac{40 \pm \sqrt{1600 - 1280 + 128s^2}}{32}$

$x = \frac{40 \pm \sqrt{320 + 128s^2}}{32}$

Simplifying the Expression

To simplify the expression, we can start by factoring out the common term $320$ from the expression under the square root:

$x = \frac{40 \pm \sqrt{320(1 + \frac{128s^2}{320})}}{32}$

$x = \frac{40 \pm \sqrt{320(1 + \frac{s^2}{2.5})}}{32}$

Finding the Side Length

To find the side length, we need to find the value of $s$ that satisfies the equation $x = \frac{40 \pm \sqrt{320(1 + \frac{s^2}{2.5})}}{32}$. Since the area of a square is equal to the square of its side length, we can set up the equation $s^2 = 16x^2 - 40x + 25$. Solving for $s$, we get:

$s = \sqrt{16x^2 - 40x + 25}$

Simplifying the Expression

To simplify the expression, we can start by factoring out the common term $16$ from the expression under the square root:

$s = \sqrt{16(x^2 - \frac{5}{2}x + \frac{25}{16})}$

$s = \sqrt{16(x - \frac{5}{4})^2}$

$s = 4|x - \frac{5}{4}|$

Finding the Side Length

To find the side length, we need to find the value of $x$ that satisfies the equation $s = 4|x - \frac{5}{4}|$. Since the side length cannot be negative, we can set up the inequality $x - \frac{5}{4} \geq 0$. Solving for $x$, we get:

$x \geq \frac{5}{4}$

Conclusion

In conclusion, the side length of the square sign is given by the expression $s = 4|x - \frac{5}{4}|$. To find the side length, we need to find the value of $x$ that satisfies the inequality $x \geq \frac{5}{4}$. The side length is equal to the absolute value of the difference between $x$ and $\frac{5}{4}$, multiplied by $4$.

Final Answer

The final answer is: $\boxed{4|x - \frac{5}{4}|}$

Introduction

In our previous article, we explored the problem of finding the side length of a square sign with an area of $16x^2 - 40x + 25$ square inches. We used algebraic techniques to solve for the side length and found that it is given by the expression $s = 4|x - \frac{5}{4}|$. In this article, we will answer some frequently asked questions about Candy's square sign.

Q: What is the area of the square sign?

A: The area of the square sign is given by the quadratic expression $16x^2 - 40x + 25$ square inches.

Q: How do I find the side length of the square sign?

A: To find the side length of the square sign, you need to find the value of $x$ that satisfies the inequality $x \geq \frac{5}{4}$. Then, you can use the expression $s = 4|x - \frac{5}{4}|$ to find the side length.

Q: What is the minimum side length of the square sign?

A: The minimum side length of the square sign is equal to the absolute value of the difference between $x$ and $\frac{5}{4}$, multiplied by $4$. Since $x \geq \frac{5}{4}$, the minimum side length is equal to $4 \cdot 0 = 0$ inches.

Q: What is the maximum side length of the square sign?

A: The maximum side length of the square sign is equal to the absolute value of the difference between $x$ and $\frac{5}{4}$, multiplied by $4$. Since $x \geq \frac{5}{4}$, the maximum side length is equal to $4 \cdot \infty = \infty$ inches.

Q: Can I use the quadratic formula to solve for the side length?

A: Yes, you can use the quadratic formula to solve for the side length. However, it is not necessary to use the quadratic formula in this case, as we can use the expression $s = 4|x - \frac{5}{4}|$ to find the side length.

Q: What is the significance of the value $\frac{5}{4}$ in the expression $s = 4|x - \frac{5}{4}|$?

A: The value $\frac{5}{4}$ represents the x-coordinate of the vertex of the parabola $16x^2 - 40x + 25$. This value is important because it represents the maximum or minimum value of the quadratic expression.

Q: Can I use the expression $s = 4|x - \frac{5}{4}|$ to find the side length of a square sign with a different area?

A: No, you cannot use the expression $s = 4|x - \frac{5}{4}|$ to find the side length of a square sign with a different area. This expression is specific to the quadratic expression $16x^2 - 40x + 25$ and cannot be used to find the side length of a square sign with a different area.

Conclusion

In conclusion, Candy's square sign is a problem that involves finding the side length of a square sign with a given area. We used algebraic techniques to solve for the side length and found that it is given by the expression $s = 4|x - \frac{5}{4}|$. We also answered some frequently asked questions about Candy's square sign.

Final Answer

The final answer is: $\boxed{4|x - \frac{5}{4}|}$