Question 99588: Number Problem. Find two consecutive positive integers such that the sum of their square is 85 Found 2 solutions by timmy1729, bucky:Answer by timmy1729(23) (Show Source):
You can put this solution on YOUR website!
Make x one of those numbers and the x+1 would be the next consecutive number.
then so and are solutions. Since a negative squared is positive, you can just use 6 and 7.
You can put this solution on YOUR website! First recognize that if x is one of the integers, the next integer is x + 1
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Their respective squares are x^2 and (x + 1)^2. But (x + 1)^2 is equal to x^2 + 2x + 1.
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Therefore, if you add their squares you get that the sum of their squares is:
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x^2 + x^2 + 2x + 1
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Note that the two x^2 terms combine and this changes the sum of the squares to:
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2x^2 + 2x + 1
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But the problem tells you that this sum equals 85. So set this polynomial equal to 85 and the
equation is then:
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2x^2 + 2x + 1 = 85
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Subtract 85 from both sides to get this equation in a standard quadratic form of:
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2x^2 + 2x - 84 = 0
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Notice that each term has 2 as a factor. Therefore, you can simplify this equation a
little by dividing all the terms on both sides by 2. This reduces the equation to:
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x^2 + x - 42 = 0
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The left side of this equation factors into (x + 7)*(x - 6) which makes the equation
become:
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(x + 7)*(x - 6) = 0
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Notice also that this equation will be true whenever one of the factors is equal to zero
because multiplication involving a zero in the left side will cause the entire left side
to equal zero which makes it equal to the right side.
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Therefore, the equation will be true if either (x + 7) = 0 or (x - 6) = 0
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Solving the first one tells you that if x = -7 that factor will be zero. But x can't be
negative because the problem says that both integers are consecutive and positive.
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Solve for the second factor (x - 6) = 0. This tells you that if x = +6 the equation will
be true. This solution looks good because x equals a positive integer.
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At the beginning we said the two integers were x and x + 1. We know that x is equal to
+6 and if we add 1 to that we get +7. So our answer is that the two consecutive and positive
integers are +6 and +7.
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Check by finding their squares, adding them, and seeing if that sum is 85.
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6^2 + 7^2 = 36 + 49 = 85
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Our answer checks.
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Hope this helps you to understand the problem and see how to work through it to get the
answer.
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