Question

The number of integer solutions of the equation (x2−10)(x2−3x−10)=1 is

Answer 1

Correct Answer: 4

For the given equation, we have:

(x^2 - 10)^(x^2 - 3x - 10) = 1 

Since any non-zero number raised to the power of 0 is 1, we can immediately see that one possible solution is when the base (x^2 - 10) is equal to 1: 

x^2 - 10 = 1 
x^2 = 11 

This gives us two solutions for x: x = √11 and x = -√11. 

Now, let's consider the case where the exponent (x^2 - 3x - 10) is equal to 0: 

x^2 - 3x - 10 = 0 

This is a quadratic equation that can be factored: 

(x - 5)(x + 2) = 0 

This gives us two more solutions: x = 5 and x = -2. 

However, we need to check whether these solutions satisfy the original equation: 

For x = √11 and x = -√11: 
(x^2 - 10)^(x^2 - 3x - 10) = (11 - 10)^0 = 1^0 = 1 

For x = 5 and x = -2: 
(x^2 - 10)^(x^2 - 3x - 10) = (25 - 10)^(25 - 3(5) - 10) = 15^0 = 1 

All four solutions satisfy the given equation, so the total number of integer solutions is 4: x = √11, x = -√11, x = 5, and x = -2.