Use the following facts. If x represents an integer, then x+1 ...

Question

Use 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 even integers is 52. 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 even integers, who some of their squares is 100. So in this case, we're looking for two unknown intruders who I can call X and Y. And the one thing that I know is that when I take the square of the first integer or X squared, and I add the square of the second integer, Y square, I get 100. But this equation isn't helpful because I have two unknown variables in this equation. So I'm going to try to relate these two variable so that I only have one variable. So in this case, two consecutive even integers is what I'm looking for. So when we think of a number line, I have an even number and then I have an odd number and then another even number. And I know that my integers are consecutive. So each even integer on a number line is two units apart. So I can say that Y is equal to my first number X plus two. And once I redefine why that way I can plug it in back into this equation. So I have X squared plus X plus two squared is equal to 100 and I can expand that exponents. I have X plus two times X plus two is equal to 100. If I multiply have X times X is X squared, X times two is two X and then I have two times X is two X and two times two is four. And let me not forget to carry down this X squared. So if I combine like terms, I have two X squared plus X squared, which is two X squared, two X plus two, X is for X plus four is equal to 100. And I'm going to move the 100 to the left hand side by subtracting by 100. So that leaves me with two X squared plus four X And positive -100 is negative Is equal to zero. I can take out a two or divide both sides by two. And when I do that two X squared divided by two is X squared four, X divided by two is two X and negative 96 divided by two is negative 48 0, divided by two is zero. And now you can see I have this quadratic equation I can solve it by or solve X by factoring. So I want two factors. And when I multiply the first term I need to get X squared. So I can do X times X gives me X squared and then I need the last terms to multiply to give me negative 48. So when I think of factors of -48, I can try 12 and 4, 12 and negative four or vice versa, negative 12 and positive four or six, negative six and positive eight or positive six and negative eight. So I'll go ahead and try positive eight and negative six. So my two factors would be X plus eight and X minus six. And I can test this out using foil. So I have X times X is X squared, X times negative six is negative six, X positive eight times X is positive eight, X and positive eight times negative six is negative 48. I combine these middle terms. I have X squared plus two, X because negative six plus eight is positive two -48. And you can see that this equation is the same as my original. So I know that these factors are good. Now I can solve for X using the factors. So I have X plus eight equals zero and X minus six equals zero when I saw for X subtract eight which gives Me X is equal to -8. And for the second factor I'll add six to both sides. So X is equal to positive six. So you can see that I have two solutions for X X could be -8. Or it could be six. But my, so that gives me my first even number and recall the second even number because X has two possible solutions. That means that there's two possible solutions for the second number. So I could also have negative eight plus two, which is negative six or six plus two, which is eight, Which means that I could have as my possible solutions negative eight and negative six or six and positive eight As my consecutive even numbers whose square is 100. And if we look at the answer choices, you can see answer choice. A also lists negative eight and negative six or six and eight. Hopefully this video helped and see you next time.

Key Concepts

Consecutive Even Integers

Algebraic Representation of Word Problems

Solving Quadratic Equations

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