Answer each question. Answer each question. Answer each question....

Question

Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. The sum of the squares of two consecutive odd integers is 202. Find the integers

Accepted Answer

Hello, everyone. In this video, we're going to be looking at this practice problem where we want to find two consecutive odd integers whose sum of their squares is 394. So in this case, we have two unknown integers, which I can call X and Y and all I know is that when I square those integers, so I have X squared plus Y squared, I get 394. But you can see in this equation, I have two variables. So I can't really solve for either one of them. So I want to try to find a way to relate the two numbers or in this case X&Y. So because it says that we have two consecutive odd integers, we can think of a number line. So we have an odd number and then an even number and then an odd number again. So you can see that between two consecutive odd numbers, there is two units. So I can say that my second number because I know it's going to be following. The first odd number is going to be the first number X plus two units. So once I redefine why? In this way, I can plug it back into this equation. So I have X squared plus X plus two squared is equal to 94. And now you can see that my equation only has X B X variable. So I'm gonna solve for X, I have X plus two times X plus two, we'll distribute or foil this. So I have X times X is X squared, X times two is two, X, two times X is two, X again and two times two is four is still have this X squared. And if I combine like terms X squared plus X squared is two, X squared, two, X plus two, X is for X plus four, go to three 94. And I'll move the 394 over to the left hand side by subtracting. She leaves me with two X squared plus four X. Now I have positive four minus 3 94 is negative 3 90 equal to zero. I can take out a two from this, both sides and I have two X squared divided by two is X squared. Four, X divided by two is two X and negative 3 90 divided by two is negative 195. The right hand side is zero, divided by two which stays zero. And now you can see I have a quadratic equation. So I want to factor it. I can try to solve for X by factoring it. So in my factoring, I want the first term to multiply to give me X squared. So X times X would give me X squared. So I'll try that, I'll try having access my first term first And then I need my second terms to multiply to give me -195. So thinking of some factors for 1 95 I have 15 and 13 and again, I need negative 1 95. So I could have negative 15 or positive 15 and negative 13. I could also have three and negative vice versa or I could have five and negative 39. So trying these factors, I'm just going to go ahead and try 15 and negative 13 and I can check this by doing foil and making sure I get back my original equation. So I have X times X is X squared, X times negative 13 is negative 13, X positive 15 times X is positive 15, X positive 15 times negative 13 is negative 1 95. If I combine my middle terms, I get back negative 13 plus 15, X is 2X. And you can see that this is indeed my original equation. So these factors are good and I can solve for X using the factors. So I have X plus 15 is equal to zero and X -13 is equal to zero, which means that X could be -15, 13 for this one. X could also be positive 13. So in this case, I have two solutions for X, it could be negative 15 and positive 13. But recall that X is only one of the integers that we're looking for. We also need the second integer which I denoted as why? So because X has two solutions, that means that why could have two solutions as well? So why could be negative 15 plus two or 13 plus two? So negative 15 plus two is negative 13 and 13 plus two is positive 15. And remember that there are two consecutive odd integers. So when I have negative 15, my second integer would be negative 13. And if I have positive 13, then my second integer could be positive 15. So this becomes my final answer for these two integers. And if we look at the answer choices, you can see that D also lists -15 and - or positive 13 and positive 15. So hopefully this video helped and see you next time.

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