For equation $(x^{2} - 5 x + 5)^{x^{2} + 4 x - 60} = 1$

Number of real value of

x = 4

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Number of real value of

x = 5

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Sum of real value of

x = 1

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Sum of real value of

x = 3

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Solution

The correct option is A Number of real value of $x = 4$
${(x^{2} - 5 x + 5)}^{x^{2} + 4 x - 60} = 1 ⟶ (1) \Rightarrow l o g {(x^{2} - 5 x + 5)}^{x^{2} + 4 x - 60} = l o g 1 (t a k i n g l o g o n b o t h s i d e) \Rightarrow (x^{2} + 4 x - 60) l o g (x^{2} - 5 x + 5) = 0 (s i n c e l o g 1 = 0) s o, (x^{2} + 4 x - 60) = 0 (0 r) l o g (x^{2} - 5 x + 5) = 0 f o r, (x^{2} + 4 x - 60) = 0 \Rightarrow b^{2} - 4 a c = 4^{2} - 4.1. (- 60) = 256 > 0 ∴ x h a s 2 d i s t i n c t r e a l r o o t s a l s o f o r, l o g (x^{2} - 5 x + 5) = 0 \Rightarrow l o g (x^{2} - 5 x + 5) = l o g 1 (s i n c e l o g 1 = 0) \Rightarrow (x^{2} - 5 x + 5) = 0 \Rightarrow b^{2} - 4 a c = {(- 5)}^{2} - 4.1.4 = 9 > 0 ∴ x h a s 2 d i s t i n c t r e a l r o o t s ∴ f o r e q u a t i o n (1), n u m b e r o f r e a l v a l u e s o f x = 4$

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

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