https://file.aiquickdraw.com/mathbot/file/1726544202388gwri63x9.jpg

A computer company makes a rectangular screen with a diagonal of 20 inches. The width of the screen is 4 inches less than its length. 

The dimensions of the computer screen are modeled by the equation x2 + (x – 4)2 = 202. What is the value of x, the length of the screen?
x = –16
x = –12
x = 12
x = 16

We are given a rectangular screen with a diagonal of 20 inches, and the width of the screen is 4 inches less than its length. To model this, we use the Pythagorean theorem, which states:

\[
\text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2
\]

From the diagram:
- The diagonal is 20 inches.
- The length is \( x \) inches.
- The width is \( x - 4 \) inches.

The equation modeling this situation is:

\[
x^2 + (x - 4)^2 = 20^2
\]

First, let's solve this equation step by step.

### Step 1: Expand the equation
\[
x^2 + (x - 4)^2 = 400
\]

Expanding \( (x - 4)^2 \):
\[
(x - 4)^2 = x^2 - 8x + 16
\]

So the equation becomes:
\[
x^2 + x^2 - 8x + 16 = 400
\]

### Step 2: Simplify the equation
Combine like terms:
\[
2x^2 - 8x + 16 = 400
\]

### Step 3: Move all terms to one side of the equation
\[
2x^2 - 8x + 16 - 400 = 0
\]

\[
2x^2 - 8x - 384 = 0
\]

### Step 4: Divide the entire equation by 2 to simplify
\[
x^2 - 4x - 192 = 0
\]

### Step 5: Solve using the quadratic formula
The quadratic formula is given by:

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

For the equation \( x^2 - 4x - 192 = 0 \), \( a = 1 \), \( b = -4 \), and \( c = -192 \).

Let's calculate this.The solutions to the quadratic equation are \( x = 16 \) and \( x = -12 \). Since the length of the screen cannot be negative, the valid solution is:

\[
x = 16
\]

Thus, the length of the screen is 16 inches.

Would you like more details or further clarification?

Here are 5 related questions for further practice:
1. How would you find the width of the screen using the value of \( x \)?
2. What would the area of the screen be, using the length and width?
3. How can the Pythagorean theorem be applied to other shapes?
4. What are some real-world applications of the quadratic formula?
5. How would the equation change if the width was a different amount less than the length?

**Tip:** When solving a quadratic equation, always check for extraneous or invalid solutions, especially in real-world problems where certain values (like negative lengths) don't make sense.

A computer company makes a rectangular screen with a diagonal of 20 inches. The width of the screen is 4 inches less than its length. The dimensions of the computer screen are modeled by the equation x^2 + (x – 4)^2 = 20^2. What is the value of x, the length of the screen?

Learn how to find the length of a computer screen with a diagonal of 20 inches, where the width is 4 inches less than its length. This problem involves using algebraic techniques, including the quadratic equation and the Pythagorean Theorem. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation