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  4. The sum of all real values of x satisfying the equation (x2 - 5x + 5)x2 + 4x -60 = 1 is

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Q. The sum of all real values of $x$ satisfying the equation $(x^2 - 5x + 5)^{x^2 + 4x -60} = 1 $ is

JEE MainJEE Main 2016Complex Numbers and Quadratic Equations

Solution:

$\left(x^{2} - 5x + 5\right)^{x^2 +4x-60} = 1 = \left(x^{2} - 5x +5 \right)^{0} $
$\Rightarrow x^{2} + 4x -60 = 0 \left[a^{x} = a^{y} \Rightarrow x=y \, \, \, if \, a \neq 1, 0 , -1\right] $
$x =- 10, 6$
& base $ x^{2} -5x +5 = 0 $ or $1$ or $-1$
If $x^2 - 5x + 5 = 0 $
image
But it will not satisfy original equation .
Hence solutions are - $10, 6, 4, 1, 2$
So, sum of solutions $= - 10 + 6 + 4 + 1 + 2 = 3$