1) The sum of a number and its reciprocal is 25/12. Find the number.
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Set up your equation:
x + 1/x = 25/12
Multiply everything by 12x to cancel off all denominators:
12x^2 + 12 = 25x
Move everything to one side:
12x^2 - 25x + 12 = 0
Factor it out:
(4x - 3) (3x - 4) = 0
Now solve for each:
4x - 3 = 0
4x = 3
x = 3/4
3x - 4 = 0
3x = 4
x = 4/3
Because addition is commutative both 3/4 and 4/3 are valid solutions.
x + 1/x = 25/12
Multiply everything by 12x to cancel off all denominators:
12x^2 + 12 = 25x
Move everything to one side:
12x^2 - 25x + 12 = 0
Factor it out:
(4x - 3) (3x - 4) = 0
Now solve for each:
4x - 3 = 0
4x = 3
x = 3/4
3x - 4 = 0
3x = 4
x = 4/3
Because addition is commutative both 3/4 and 4/3 are valid solutions.
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x+1/x=25/12
Multiply both sides by x, and put into standard polynomial form
x^2+1=25x/12
x^2-25x/12+1=0
x^2+2(-25/24)x+1=0
x^2=2(-25/24)x+(25/24)^2=-1+(25/24)^2=…
(x-25/24)^2=(7/24)^2
x-25/24=7/24
x=25/24+7/24=32/24=4/3
Ans: x=4/3
Multiply both sides by x, and put into standard polynomial form
x^2+1=25x/12
x^2-25x/12+1=0
x^2+2(-25/24)x+1=0
x^2=2(-25/24)x+(25/24)^2=-1+(25/24)^2=…
(x-25/24)^2=(7/24)^2
x-25/24=7/24
x=25/24+7/24=32/24=4/3
Ans: x=4/3
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x+1/x = 25/12 => 12x^2-25x+12 = 0. Solving this quadratic equation we get x = 4/3 or 3/4.
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The reciprocal of x is 1/x, therefore the reciprocal of 25/12 is 12/25. Just flip it!
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x + (1/x) = 25/12
12(x+1) =25x
12x^2-25x+12 = 0
X= 4/3
12(x+1) =25x
12x^2-25x+12 = 0
X= 4/3