SOLUTION: Find three consecutive integers that the sum of twice the largest and the fourth power of the smallest is equal to the square of the remaining integer increased by 75.
First int
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Question 265600: Find three consecutive integers that the sum of twice the largest and the fourth power of the smallest is equal to the square of the remaining integer increased by 75.
First int __,__
Second int ___,___
Third int___,___
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Let 
represent the smallest of the three consecutive integers. Then next larger one must then be 
, and the largest must be 
.
The fourth power of the smallest is: 
Twice the largest is: \ =\ 2x\ +\ 4)
The middle one squared increased by 75 is then: ^2\,+\,75\ =\ x^2\ +\ 2x\ +\ 1\ +\ 75\ =\ x^2\ +\ 2x\ +\ 76)
Putting it all together:
Collect like terms:
Let 
, then:
Factor the quadratic:
(u\,-\,9)\ =\ 0)
Hence:
or
Which is to say:
which has no real number solutions
or
which means
or
Hence the three integers are either 3, 4, and 5 or -3, -2, and -1.
Check:
\ =\ 91)
Answer checks.
^4\ +\ 2(-1)\ =\ 79)
^2\ +\ 75\ =\ 79)
Answer checks.
John
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