1. **Problem statement:** A tablet screen measures 8 inches by 5 inches, and it has a border of uniform thickness $x$ around it. We need to find expressions and solve for $x$ based on the total area including the border. 2. **Part a: Expression for total area including the border** The screen dimensions are 8 inches by 5 inches. The border adds $x$ inches on all sides, so the total width is $8 + 2x$ and the total height is $5 + 2x$. The total area $A$ including the border is: $$A = (8 + 2x)(5 + 2x)$$ 3. **Part b: Equation for total area equal to 50.3125** We set the total area equal to 50.3125: $$ (8 + 2x)(5 + 2x) = 50.3125 $$ A solution to this equation means the thickness $x$ of the border that makes the total tablet area exactly 50.3125 square inches. 4. **Part c: Solve the equation** First, expand the left side: $$ (8 + 2x)(5 + 2x) = 8 \times 5 + 8 \times 2x + 2x \times 5 + 2x \times 2x = 40 + 16x + 10x + 4x^2 $$ Combine like terms: $$ 4x^2 + 26x + 40 = 50.3125 $$ Subtract 50.3125 from both sides: $$ 4x^2 + 26x + 40 - 50.3125 = 0 $$ $$ 4x^2 + 26x - 10.3125 = 0 $$ Divide entire equation by 2 to simplify: $$ \cancel{4}x^2/\cancel{2} + \cancel{26}x/\cancel{2} - 10.3125/2 = 0 $$ $$ 2x^2 + 13x - 5.15625 = 0 $$ Use the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=2$, $b=13$, $c=-5.15625$. Calculate discriminant: $$ b^2 - 4ac = 13^2 - 4 \times 2 \times (-5.15625) = 169 + 41.25 = 210.25 $$ Square root: $$ \sqrt{210.25} = 14.5 $$ Calculate solutions: $$ x = \frac{-13 \pm 14.5}{2 \times 2} = \frac{-13 \pm 14.5}{4} $$ Two possible values: $$ x_1 = \frac{-13 + 14.5}{4} = \frac{1.5}{4} = 0.375 $$ $$ x_2 = \frac{-13 - 14.5}{4} = \frac{-27.5}{4} = -6.875 $$ Since thickness cannot be negative, the border thickness is: $$ \boxed{0.375 \text{ inches}} $$