Question

The sum of values of\[x\]satisfying the equation\[{\left( {31 + 8\sqrt {15} } \right)^{{x^2} - 3}} + 1 = {\left( {32 + 8\sqrt {15} } \right)^{{x^2} - 3}}\]is
A. \[\sqrt 3 \]
B. \[2\]
C. \[0\]
D. None of the above

Answer 1

Hint: The given problem involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We need to find what sum value \[x\]satisfies the given equation. Also, we need to know the arithmetic operations with the involvement of power terms and square root terms.

Complete step by step solution:
The given equation is shown below,
\[{\left( {31 + 8\sqrt {15} } \right)^{{x^2} - 3}} + 1 = {\left( {32 + 8\sqrt {15} } \right)^{{x^2} - 3}} \to \left( 1 \right)\]
The above equation satisfied only when \[{x^2} - 3 = 1\]
We know that, if we put\[1\]as a power of any term, the value of the term wouldn’t change. That is, \[{n^1} = 1\]
By substituting \[{x^2} - 3 = 1\]in the equation\[\left( 1 \right)\], we get
\[\left( 1 \right) \to {\left( {31 + 8\sqrt {15} } \right)^{{x^2} - 3}} + 1 = {\left( {32 + 8\sqrt {15} } \right)^{{x^2} - 3}}\]
\[
  {\left( {31 + 8\sqrt {15} } \right)^1} + 1 = {\left( {32 + 8\sqrt {15} } \right)^1} \\
  31 + 1 + 8\sqrt {15} = 32 + 8\sqrt {15} \\
  32 + 8\sqrt {15} = 32 + 8\sqrt {15} \\
 \]
So, we get LHS is equal to RHS, the given equation satisfied.
Let’s find\[x\]value from\[{x^2} - 3 = 1\]
We have,
\[{x^2} - 3 = 1\]
Let’s move\[ - 3\]from LHS to RHS, we get
\[{x^2} = 3 + 1\]
\[{x^2} = 4\]
Take square root on both sides
\[
  \sqrt {{x^2}} = \sqrt 4 \\
  x = \pm 2 \\
 \]
So, the sum of the\[x\]value, which satisfies the given equation\[ = - 2 + 2 = 0\]
So, the final answer is,
The sum of the \[x\] value, which satisfies the given equation is equal to \[0\].

So, the correct answer is “Option C”.

Note: This question describes the arithmetic operations like addition/ subtraction/ multiplication/ division. Note that when we square the negative term the answer becomes a positive term. Also, note that every whole number has\[1\]in the power. When we multiply different sign terms, we would follow the following things,
1. When a negative term is multiplied with the negative term the final answer will be a positive term.
2. When a positive term is multiplied with a positive term the final answer will be a positive term
3. When we multiply a negative number with the positive number the final answer will be a negative number.