+619 \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 $ab + ad + bc + cd$?
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 !!.
