A garden in the shape of a rectangle is surrounded through a walkway of uniform width. The dimensions of the garden only are 35 by 24. The field of the garden and the walkway together is 1,530 square feet. What is the width of the walkway in feet?
Let x = the width of the walkway. Since the width of the garden just is 24, the width of the garden and the walkway together is x + x + 24 or 2x + 24. Because the length of the garden only is 35, the length of the garden and the walkway together is x + x + 35 or 2x + 35. Area of a rectangle is length times width, so multiply the expressions together and set the result equal to the total area of 1,530 square feet: (2x + 24)(2x + 35) = 1,530. Multiply the binomials by using the distributive property: 4x^{2} + 70x + 48x + 840 = 1,530. Combine such as terms: 4x^{2} + 118x + 840 = 1,530. Subtract 1,530 from both sides: 4x^{2} + 118x + 840 - 1,530 = 1,530 - 1,530; simplify: 4x2 + 118x - 690 = 0. Factor the trinomial fully: 2(2x^{2} + 59x - 345) = 0; 2(2x + 69)(x - 5) = 0. Set each factor equal to zero and solve: 2 ≠ 0 or 2x + 69 = 0 or x - 5 = 0; x = -34.5 or x = 5. Reject the negative solution since you will not have a negative width. The width is 5 feet.