Question

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? \\(\bigcirc\\) $x = -16$ \\(\bigcirc\\) $x = -12$ \\(\bigcirc\\) $x = 12$ \\(\bigcirc\\) $x = 16$ (image: a rectangle with length x in., width (x - 4) in., and diagonal 20 in. not drawn to scale)

Explanation:

Step1: Expand the equation

Given \(x^{2}+(x - 4)^{2}=20^{2}\), expand \((x - 4)^{2}\) using the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\). So \((x - 4)^{2}=x^{2}-8x + 16\) and \(20^{2}=400\). The equation becomes \(x^{2}+x^{2}-8x + 16 = 400\).

Step2: Simplify the equation

Combine like terms: \(2x^{2}-8x + 16-400 = 0\), which simplifies to \(2x^{2}-8x - 384 = 0\). Divide the entire equation by 2: \(x^{2}-4x - 192 = 0\).

Step3: Factor the quadratic equation

Factor \(x^{2}-4x - 192\). We need two numbers that multiply to -192 and add to -4. The numbers are -16 and 12. So the equation factors as \((x - 16)(x + 12)=0\).

Step4: Solve for x

Set each factor equal to zero: \(x - 16 = 0\) or \(x + 12 = 0\). This gives \(x = 16\) or \(x=-12\). Since length cannot be negative, we discard \(x=-12\).

Answer:

\(x = 16\) (corresponding to the option D. \(x = 16\) if options are labeled as A, B, C, D with D being \(x = 16\); here the correct option is the one with \(x = 16\))