Solving For The Dimensions Of A Computer Screen A Mathematical Exploration
Hey there, math enthusiasts! Today, we're diving into a fascinating problem involving a rectangular computer screen. A certain computer company is crafting these screens with a diagonal that measures 20 inches. Now, here's the kicker: the width of the screen is 4 inches shorter than its length. Our mission, should we choose to accept it, is to unravel the dimensions of this screen using a nifty equation.
The Mathematical Model
The dimensions of this computer screen are cleverly modeled by the equation:
x² + (x - 4)² = 20²
Where 'x' represents the length of the screen. Let's break down this equation and understand its origins. Picture a rectangle, our computer screen. The diagonal acts as the hypotenuse of a right-angled triangle, with the length and width forming the other two sides. The Pythagorean theorem, a cornerstone of geometry, states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In our case, the diagonal (hypotenuse) is 20 inches, the length is 'x', and the width is 'x - 4'. Applying the Pythagorean theorem, we get:
x² + (x - 4)² = 20²
This equation beautifully captures the relationship between the screen's dimensions and its diagonal. Our task now is to solve this equation for 'x', which will reveal the length of the screen. Once we have the length, finding the width is a breeze – we simply subtract 4 inches.
Cracking the Equation
Alright, let's roll up our sleeves and solve this equation. First, we need to expand the equation and simplify it. Remember, the goal is to isolate 'x' and find its value. Here's how we proceed:
-
Expand (x - 4)²: This means multiplying (x - 4) by itself. Using the FOIL (First, Outer, Inner, Last) method or the binomial theorem, we get:
(x - 4)² = x² - 8x + 16 -
Substitute back into the original equation: Now, we replace (x - 4)² in our original equation with its expanded form:
x² + (x² - 8x + 16) = 20² -
Simplify: Let's combine like terms and simplify the equation:
2x² - 8x + 16 = 400 -
Rearrange into a quadratic equation: To solve for 'x', we need to get the equation into the standard quadratic form, which is ax² + bx + c = 0. Subtract 400 from both sides:
2x² - 8x - 384 = 0 -
Divide by a common factor: Notice that all the coefficients are divisible by 2. Dividing the entire equation by 2 simplifies it:
x² - 4x - 192 = 0
Now we have a much cleaner quadratic equation to solve! There are a couple of ways we can tackle this: factoring or using the quadratic formula. Let's explore both.
Factoring the Quadratic
Factoring involves finding two numbers that multiply to give the constant term (-192) and add up to the coefficient of the x term (-4). This might sound tricky, but with a little thought, we can crack it. We need to think of factor pairs of 192 and see which pair has a difference of 4.
The factors of 192 are: 1 and 192, 2 and 96, 3 and 64, 4 and 48, 6 and 32, 8 and 24, 12 and 16. Bingo! 12 and 16 have a difference of 4. Since we need a -4, we'll use -16 and +12.
So, we can rewrite our quadratic equation as:
(x - 16)(x + 12) = 0
For this product to be zero, one or both of the factors must be zero. This gives us two possible solutions for x:
x - 16 = 0 => x = 16
x + 12 = 0 => x = -12
Now, hold on a second! We're dealing with the dimensions of a screen, and dimensions can't be negative. So, we discard the solution x = -12. This leaves us with x = 16.
Using the Quadratic Formula
If factoring feels like a puzzle you can't quite solve, the quadratic formula is your trusty backup. It's a foolproof way to find the roots of any quadratic equation in the form ax² + bx + c = 0. The formula is:
x = [-b ± √(b² - 4ac)] / 2a
In our equation, x² - 4x - 192 = 0, we have a = 1, b = -4, and c = -192. Let's plug these values into the formula:
x = [4 ± √((-4)² - 4 * 1 * -192)] / (2 * 1)
x = [4 ± √(16 + 768)] / 2
x = [4 ± √784] / 2
x = [4 ± 28] / 2
This gives us two solutions:
x = (4 + 28) / 2 = 32 / 2 = 16
x = (4 - 28) / 2 = -24 / 2 = -12
Just like with factoring, we discard the negative solution x = -12, leaving us with x = 16.
The Screen's Dimensions Unveiled
We've cracked the code! The value of x, which represents the length of the screen, is 16 inches. Now, to find the width, we simply subtract 4 inches from the length:
Width = x - 4 = 16 - 4 = 12 inches
So, the dimensions of the computer screen are 16 inches in length and 12 inches in width. We've successfully navigated the mathematical landscape and emerged with the solution! This journey showcases the power of the Pythagorean theorem and quadratic equations in solving real-world problems. Remember guys, math is not just about numbers; it's about understanding the relationships that govern our world.
What is the Value of x?
So, the big question we've been tackling is: what is the value of x that represents the length of the screen in the equation x² + (x - 4)² = 20²? We've journeyed through expanding the equation, simplifying it into a quadratic form, and then employing two different methods – factoring and the quadratic formula – to arrive at the answer. Both methods led us to the same conclusion: x = 16.
Therefore, the value of x that satisfies the equation and represents the length of the computer screen is 16 inches. This wasn't just about crunching numbers; it was about applying mathematical principles to a practical scenario and understanding how equations can model real-world relationships. We've seen how the Pythagorean theorem lays the foundation for this problem, and how solving quadratic equations helps us find the specific dimensions.
Wrapping Up
This problem is a fantastic example of how math pops up in unexpected places, even in the design of our gadgets! By using our knowledge of geometry and algebra, we were able to determine the dimensions of the computer screen. Remember, math is a powerful tool that helps us understand and interact with the world around us. So, keep exploring, keep questioning, and keep solving! Who knows what mathematical mysteries you'll unravel next?