Question

The number of integer solutions of the equation $(x^2−10)^{(x2−3x−10)}=1$ is

Answer 1

The correct answer is: 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 $x: x = \sqrt{11}\space{and}\space x=-\sqrt{11}$.
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 $x=\sqrt{11}\space{and}\space x=-\sqrt{11}$:
$(x^2-10)^{(x^2-3x-10)}=(11-10)^0=1^0=1$
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: $x = \sqrt{11}, x = -\sqrt{11}, x = 5$, and x = -2.