Question 175265: Please help: i just can't figure these two questions out:
#1 Find two negative consecutive integers such that the square of the larger diminished by five times the smaller is equal to 71.
#2 find three consecutive odd integers such that the square of the second integer exceeds the largest integer by 10.
thank you - desperatly seeking help :)
Answer by gonzo(654) (Show Source):
You can put this solution on YOUR website! problem number 1:
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let x = first negative integer
let x + 1 = second negative integer.
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the problem states that the square of the larger minus 5 times the smaller is equal to 71.
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this means that
(x+1)^2 - 5x = 71
which means that
x^2 + 2x + 1 - 5x = 71
which means that
x^2 - 3x + 1 = 71
which means that
x^2 - 3x - 70 = 0
which means that
(x-10) * (x+7) = 0
which means that
x = 10
or
x = -7
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since the number is negative, x = -7 is the only applicable answer.
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to prove, substitute in original equation.
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the numbers are x and x+1 which means that the numbers are -7 and -6.
-6 is the larger number.
square of the larger number minus 5 times the smaller number is 71.
(-6)^2 - (5*(-7)) = 36 - (-35) = 36 + 35 = 71.
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x = -7 checks out ok.
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answer is the 2 negative consecutive numbers are -7 and -6.
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problem number 2:
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let x = one of the negative numbers,
then x + 2 = next negative number, and x + 4 = next negative number.
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largest negative number is x + 4
middle negative number is x + 2
smallest negative number is x
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problem states that the square of the middle negative number is greater than the largest negative number by 10.
this means that (x+2)^2 = (x+4) + 10
which means that
x^2 + 4x + 4 = x + 14
which means that
x^2 + 3x - 10 = 0
which means that
(x+5) * (x-2) = 0
which means that
x = -5
or
x = 2
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since the number has to be negative, then x = -5 is the only applicable answer to the problem.
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if x = -5, then
x + 2 = -3, and
x + 4 = -1
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largest number is -1
middle number is -3
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square of second number is 9
largest number + 10 is 9
square of second number equals the largest number plus 10 which satisfies the problem statement.
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answer is the 3 negative consecutive numbers are:
-5, -3, -1
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