Problem: "A 35- by 16-ft rectangular swimming pool is surrounded by a walkway of uniform width. If the total area of the walkway is 280 ft^2, how wide is the walkway?"
Now, I understand most of this, up until a snag at the very end. Here is the work I did so far:
I know you subtract the measurements of the swimming pool from the measurements of the outer rectangle, containing both the swimming pool and the walkway surrounding it.
The length of the outer rectangle is given by the expression (2x + 35), and its width is given by the expression (2x + 16). The area of the swimming pool is 560 ft^2, so we get the following equation:
(2x + 35)(2x + 16) - 560 = 280
FOILing, I was able to get it to…
4x^2 + 32x + 70x + 560 - 560 = 280
Simplified to…
4x^2 + 102x = 280
Subtracting 280 from both sides, I got…
4x^2 + 102x - 280 = 0
Dividing everything on both sides by 2, I got…
2x^2 + 51x - 140 = 0
Now here's the kicker. The next step is to factor this quadratic equation. However, there is a little issue that confuses me. Obviously, 2x^2 can be factored as being the product of 2x and x, but -140 has so many different factors that I don't know which two I am supposed to choose when factoring. I did 14 and -10, with the result that the quadratic equation was factored into the form (x + 14)(2x - 10) = 0. I then proceeded to set each term in each set of parentheses equal to 0 and solve for x, but the computer program said it was incorrect. When I gave up, unable to reason why my answer was incorrect, and asked it to show me the correct answer, it turned out that they had used different factors of -140 besides -10 and 14.
I suppose, if I had unlimited tries on the problem, I could go through all of the factors of -140 and plug them all in, and see if the computer program accepts them, but I cannot do that, as I can only get the answer wrong twice before I lose all points for that question.
So does anyone know how, in a problem of this nature, I am supposed to determine which factors of a number I am supposed to use and put in parentheses when factoring a quadratic equation, in a case like this, when the number has so many different factors to choose from?
Thank you very much.
Now, I understand most of this, up until a snag at the very end. Here is the work I did so far:
I know you subtract the measurements of the swimming pool from the measurements of the outer rectangle, containing both the swimming pool and the walkway surrounding it.
The length of the outer rectangle is given by the expression (2x + 35), and its width is given by the expression (2x + 16). The area of the swimming pool is 560 ft^2, so we get the following equation:
(2x + 35)(2x + 16) - 560 = 280
FOILing, I was able to get it to…
4x^2 + 32x + 70x + 560 - 560 = 280
Simplified to…
4x^2 + 102x = 280
Subtracting 280 from both sides, I got…
4x^2 + 102x - 280 = 0
Dividing everything on both sides by 2, I got…
2x^2 + 51x - 140 = 0
Now here's the kicker. The next step is to factor this quadratic equation. However, there is a little issue that confuses me. Obviously, 2x^2 can be factored as being the product of 2x and x, but -140 has so many different factors that I don't know which two I am supposed to choose when factoring. I did 14 and -10, with the result that the quadratic equation was factored into the form (x + 14)(2x - 10) = 0. I then proceeded to set each term in each set of parentheses equal to 0 and solve for x, but the computer program said it was incorrect. When I gave up, unable to reason why my answer was incorrect, and asked it to show me the correct answer, it turned out that they had used different factors of -140 besides -10 and 14.
I suppose, if I had unlimited tries on the problem, I could go through all of the factors of -140 and plug them all in, and see if the computer program accepts them, but I cannot do that, as I can only get the answer wrong twice before I lose all points for that question.
So does anyone know how, in a problem of this nature, I am supposed to determine which factors of a number I am supposed to use and put in parentheses when factoring a quadratic equation, in a case like this, when the number has so many different factors to choose from?
Thank you very much.