SOLUTION: If sqrt(5)= 2+ 1/a, show that a= 4+1/a

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Question 1118033: If sqrt(5)= 2+ 1/a, show that a= 4+1/a
Found 4 solutions by josgarithmetic, MathTherapy, ikleyn, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
If sqrt(5)= 2+ 1/a, show that a= 4+1/a
matrix%281%2C3%2C+sqrt%285%29%2C+%22=%22%2C+2+%2B+1%2Fa%29
matrix%281%2C3%2C+a+%2A+sqrt%285%29%2C+%22=%22%2C+2a+%2B+1%29 --------- Multiplying by LCD, a
matrix%281%2C3%2C+5a%5E2%2C+%22=%22%2C+4a%5E2+%2B+4a+%2B+1+%29 ---- Squaring each side
matrix%281%2C3%2C+a%5E2%2C+%22=%22%2C+4a+%2B+1%29
highlight_green%28matrix%281%2C3%2C+a%2C+%22=%22%2C+4+%2B+1%2Fa%29%29 ------- Dividing by a
OR

matrix%281%2C3%2C+sqrt%285%29%2C+%22=%22%2C+2+%2B+1%2Fa%29
matrix%281%2C3%2C+5%2C+%22=%22%2C+4+%2B+4%2Fa+%2B+1%2Fa%5E2%29 ---- Squaring each side
matrix%281%2C3%2C+1%2C+%22=%22%2C+4%2Fa+%2B+1%2Fa%5E2%29
highlight_green%28matrix%281%2C3%2C+a%2C+%22=%22%2C+4+%2B+1%2Fa%29%29 ------- Multiplying by a

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.

If sqrt(5)= 2+ 1/a, show that a= 4+1/a
~~~~~~~~~~~~~~~~~~~~~~~~ 


If sqrt%285%29 = 2 + 1%2Fa,  then  1%2Fa = sqrt%285%29-2     (1)


It implies   a = 1%2F%28sqrt%285%29-2%29    (2)


Rationalize the denominator in the right side of (2):


a = 1%2F%28sqrt%285%29-2%29 = 1%2F%28sqrt%285%29-2%29.%28sqrt%285%29%2B2%29%2F%28sqrt%285%29%2B2%29 = %28sqrt%285%29%2B2%29%2F%28%28sqrt%285%29%29%5E2+-2%5E2%29 = %28sqrt%285%29%2B2%29%2F%285-4%29 = sqrt%285%29%2B2.   (3)


Now in the formula  a = 4 + 1%2Fa    (*)


    the left  side is  a = sqrt%285%29%2B2    (according to (3))


    the right side is  4 + 1%2Fa =  (now replace 1%2Fa by sqrt%285%29-2, based on (1), to get )  = 4 + %28sqrt%285%29-2%29 = sqrt%285%29%2B2.


Thus the both sides of (*) are the same.


The statement is proved.


Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


(1) Rewrite the given equation as 1%2Fa+=+sqrt%285%29-2, and rewrite the equation to be shown as a-1%2Fa+=+4

(2) Solve the given equation to find a.

1%2Fa+=+sqrt%285%29-2
a+=+1%2F%28sqrt%285%29-2%29+=+%28sqrt%285%29%2B2%29%2F%285-4%29+=+sqrt%285%29%2B2

(3) a-1%2Fa+=+%28sqrt%285%29%2B2%29-%28sqrt%285%29-2%29+=+4