SOLUTION: a mini-TV screen has a diagonal of 6.5 inches. the length of the screen is 3.5 inches longer than the width. what are the actual dimensions of the screen. now with this I thought

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: a mini-TV screen has a diagonal of 6.5 inches. the length of the screen is 3.5 inches longer than the width. what are the actual dimensions of the screen. now with this I thought      Log On


   

Question 187398: a mini-TV screen has a diagonal of 6.5 inches. the length of the screen is 3.5 inches longer than the width. what are the actual dimensions of the screen.
now with this I thought I would try and use the Pothagorean therom. which would be a squared + b squared = c squared. so far i have gotten that 6.5 squared is 42.25
Found 2 solutions by Alan3354, solver91311:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
a mini-TV screen has a diagonal of 6.5 inches. the length of the screen is 3.5 inches longer than the width. what are the actual dimensions of the screen.
now with this I thought I would try and use the Pothagorean therom. which would be a squared + b squared = c squared. so far i have gotten that 6.5 squared is 42.25
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That's a part of it.
w is the width, so the length is w+3.5 (usually TVs are width and height, but whatever)
w%5E2+%2B+%28w%2B3.5%29%5E2+=+42.25 Pythagorean theorem
w%5E2+%2B+w%5E2+%2B+7w+%2B+12.25+=+42.25
2w%5E2+%2B+7w+-+30+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B7x%2B-30+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%287%29%5E2-4%2A2%2A-30=289.

Discriminant d=289 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-7%2B-sqrt%28+289+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%287%29%2Bsqrt%28+289+%29%29%2F2%5C2+=+2.5
x%5B2%5D+=+%28-%287%29-sqrt%28+289+%29%29%2F2%5C2+=+-6

Quadratic expression 2x%5E2%2B7x%2B-30 can be factored:
2x%5E2%2B7x%2B-30+=+%28x-2.5%29%2A%28x--6%29
Again, the answer is: 2.5, -6. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B7%2Ax%2B-30+%29

w = -6, 2.5 Ignore the -6
w = 2.5, length = 6

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Your thought about the Pythagorean Theorem was a very good thought indeed because that is the only way to solve this problem. Instead of a and b for the measures of the legs of your triangle, let's use l and w because we are talking about the sides of a rectangle after all.

So:



shows the relationship between the lengths of the sides of the rectangle with the rectangle's diagonal. By itself, that doesn't do much good because there are an infinite number of pairs of l and w values that would make this true.

Fortunately, we are also given another relationship between the length and the width of the rectangle, namely that the length is 3.5 inches longer than the width, so:



Knowing that, we can substitute w + 3.5 anywhere we see an l, so let's put it into the Pythagorean equation:



Expand using FOIL:



Collect terms and add to both sides:



Notice that you have the difference of two squares on the right, so:





Put into standard form by adding -30 to both sides:



And finally, solve the quadratic which very conveniently factors:



The Zero Product Rule gives us:



or



Notice that one of the roots of the quadratic is negative. Since it is absurd to have a negative value for the measure of length of something, we can exclude that root as being extraneous -- having been introduced by the act of squaring the variable in the process of solving the problem. The other root, the positive one, is the desired answer for the measure of the width. And the value of the measure of the length comes from simply adding 3.5 inches to the calculated width, or 2.5 + 3.5 = 6.

Check the answer:







Answer checks.

John