+0  
 
0
92
1
avatar+628 

Find the positive solution to \(\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0\)

 

Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including 1and itself. Find n/75

supermanaccz  Sep 21, 2018

Best Answer 

 #1
avatar+970 
+3

1. 

 

Let \(a = x^2 - 10x - 29\)


\(\frac{1}{a} + \frac{1}{a - 16} - \frac{2}{a - 40} = 0.\\  (a - 16)(a - 40) + a(a - 40) - 2(a)(a - 16) = 0 \\ -64a + 40 \times 16 = 0, \Rightarrow a = 10. \\\)

 

\(10 = x^2 - 10x - 29 \Longleftrightarrow 0 = (x - 13)(x + 3)\) The positive root is \(\boxed{13}.\)

 

2. 

 

https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems/Problem_5

 

I hope this helped,

 

Gavin

GYanggg  Sep 21, 2018
edited by GYanggg  Sep 21, 2018
 #1
avatar+970 
+3
Best Answer

1. 

 

Let \(a = x^2 - 10x - 29\)


\(\frac{1}{a} + \frac{1}{a - 16} - \frac{2}{a - 40} = 0.\\  (a - 16)(a - 40) + a(a - 40) - 2(a)(a - 16) = 0 \\ -64a + 40 \times 16 = 0, \Rightarrow a = 10. \\\)

 

\(10 = x^2 - 10x - 29 \Longleftrightarrow 0 = (x - 13)(x + 3)\) The positive root is \(\boxed{13}.\)

 

2. 

 

https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems/Problem_5

 

I hope this helped,

 

Gavin

GYanggg  Sep 21, 2018
edited by GYanggg  Sep 21, 2018

8 Online Users

avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.