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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}{

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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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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
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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  .........   Nov 14, 2017