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\The two values of $x$ that satisfy the equation $35x^2 - 51x + 18 = 0$ can be written as simplified fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$, where $a$, $b$, $c$, and $d$ are all positive integers. What is the value of $ab + ad + bc + cd$?

 Nov 14, 2017
 #1
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+1

I will take a crack at it !!

 

Solve for x:
35 x^2 - 51 x + 18 = 0

The left hand side factors into a product with two terms:
(5 x - 3) (7 x - 6) = 0

Split into two equations:
5 x - 3 = 0 or 7 x - 6 = 0

Add 3 to both sides:
5 x = 3 or 7 x - 6 = 0

Divide both sides by 5:
x = 3/5 or 7 x - 6 = 0

Add 6 to both sides:
x = 3/5 or 7 x = 6

Divide both sides by 7:
x = 3/5       or          x = 6/7   So now how do you assign these 4 integers to a,b,c,d? How about: a=3, b=5, c=6, d=7. So (3*5) + (3*7) + (5*6) + (6*7) =108 ??. Check it please !!.

 Nov 14, 2017
 #2
avatar+98196 
+2

35x^2 - 51x + 18 = 0

 

The solutions are  

x= 3/5  and x = 6/7

 

Letting  a = 3, b= 5, c = 6 and d = 7   we have that

 

a (b + d)  + c ( b + d)  =

 

(a + c) (b + d) =

 

( 9) ( 12)   =  108

 

 

P.S. -  Letting  a = 6, b = 7, c = 3 and d = 5   produces the same result  .........

 

cool cool cool

 Nov 14, 2017

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