Try this beautiful problem from Singapore Math Olympiad, 2012, Junior Senior based on Quadratic Equation.
Consider the equation
\(\sqrt {3x^2 - 8x + 1} + \sqrt {9x^2 - 24x - 8}\) = 3.
Quadratic Function
Analysis of Number
Root of Equation
Answer: 9
Singapore Mathematics Olympiad
Challenges and Thrills - Pre - College Mathematics
As the first hint we can assume :
y =\(3x^2 - 8x + 1 \) then the equation becomes
y + \(\sqrt {3y^2 -11}\) = 3.
Lets try to do the rest of the sum ....................
If we are still stuck after the first hint we can say :
Then \( \sqrt {3y^2 - 11}\) = 3 - y.
Lets square the both sides , we have \( 3 y^2 - 11 = 9 -6y + y^2 \) ,
Then y = 2 or y = -5
Now solve \( 3x^2 - 8x +1 = 2^2 \)
Then x =3 and \(x = - 3^{-1}\)
Hence k = \(\frac {3}{3^{-1}}\) = 9 (Answer )
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