```Question 572426
The fact that you have had some trigonometry tells me that you are well beyond algebra I. Therefore, I'm going to tell you what to do, set it up for you, give you the answers that I got, and leave you with the responsibility for doing the calculations and checking the results.
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I'll begin by telling you that there are two answers to this problem. Now let's discuss how we can determine this. We are going to use the Pythagorean theorem to solve for x. We will start with the knowledge that the three sides of this right triangle are x-5, x+3, and 15.
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First case, let's assume that the side x-5 is the hypotenuse. That means that when you square it, it will equal the sum of the squares of the other two sides. Therefore, we can write the equation:
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{{{(x-5)^2 = (x+3)^2 + 15^2}}}
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Doing the squares results in:
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{{{x^2-10x+25 = x^2 +6x +9 + 225}}}
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Solve this for x (note that the two x^2 terms cancel out) and we get that:
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{{{x = -209/16}}} and this is equivalent to {{{x = -13.065}}}
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Now we substitute this value of x into the expressions for the sides and we find that the sides are -18.065, -10.065, and 15. But negative lengths make no sense. So we now know that x-5 is not the hypotenuse.
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Second, let's assume that x+3 is the hypotenuse. So the Pythagorean theorem results in:
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{{{(x+3)^2 = (x-5)^2 + 15^2}}}
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Squaring terms results in:
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{{{x^2 +6x+9 = x^2-10x+25 + 225}}}
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When we solve this for x we get:
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{{{x = 241/16}}} which is equivalent to {{{x = 15.0625}}}
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Substituting this value for x into the expressions for the sides we find that the three sides are 10.0625, 18.0625, and 15. This looks good, so we can say that x can equal 15.0625.
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Finally, we can assume that 15 is the hypotenuse. This makes the Pythagorean equation become:
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{{{15^2 = (x-5)^2 +(x+3)^2}}}
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Squaring this out we get:
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{{{225 = x^2 -10x +25 + x^2 +6x +9}}}
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We gather and combine the terms as appropriate and write the equation in standard quadratic form. Then we solve for x using the quadratic formula to find that x = 10.8234413522 or x = -8.82344135219.
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Obviously if we substitute the -8.82344135219 into the expressions for the sides, we again get sides that have negative lengths. That makes no sense, so we discard that possibility by saying that x does not equal a negative value.
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But the value x = 10.8234413522 looks OK. If we substitute that into the expressions for the sides we get that the three sides are 13.8234413522, 5.8234413522, and 15. This is a valid possibility.
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So the two good answers for x are 15.0625 and 10.8234413522. In one case the side x+3 is the hypotenuse, and in the other case the side 15 is the hypotenuse.
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Hope this helps you to see how to work the problem. You can check the above answers by taking the two sets of numerical values that we found for the sides, and substituting those values into the Pythagorean formula to ensure that the square of the longest side in either case equals the sum of the squares of the other two sides.
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