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Step-by-Step Solution
Step 1: Recognize the Expression
We need to find the coefficients of $x^4$ and $x^2$ in the expression
$ \bigl(x + \sqrt{x^2 - 1}\bigr)^6 + \bigl(x - \sqrt{x^2 - 1}\bigr)^6.$
Let $a = \sqrt{x^2 - 1}.$ Then our expression becomes $(x + a)^6 + (x - a)^6.$
Step 2: Use the Binomial Expansion Identity
Recall that
$$(x + a)^n + (x - a)^n =
\begin{cases}
2 \bigl[ \text{even-index terms} \bigr] & \text{if } n \text{ is even},\\
2 \bigl[ \text{odd-index terms} \bigr] & \text{if } n \text{ is odd}.
\end{cases}$$
Since $n=6$ (an even integer), we only keep the terms where the binomial index $k$ is even (i.e., $k=0,2,4,6$), and each of those terms doubles.
Step 3: Write Out the Relevant Terms
The general binomial term in $(x+a)^6$ is
$${}^6C_{k}\,x^{6-k}\,a^{k}.$$
For the sum $(x+a)^6 + (x-a)^6,$ the terms with $k$ even survive. Hence we consider $k=0,2,4,6$ and multiply each by 2:
$$
(x+a)^6 + (x-a)^6
= 2 \Bigl[
{}^6C_0\,x^6\,a^0
+ {}^6C_2\,x^4\,a^2
+ {}^6C_4\,x^2\,a^4
+ {}^6C_6\,x^0\,a^6
\Bigr].
$$
Step 4: Substitute $a = \sqrt{x^2 - 1}$
We have $a^2 = x^2 - 1,$ $a^4 = (x^2 - 1)^2$, and so on. Substituting:
$$
a^2 = x^2 - 1,\quad
a^4 = (x^2 - 1)^2,\quad
a^6 = (x^2 - 1)^3.
$$
Then the expression becomes:
$$
2 \Bigl[
{}^6C_0\,x^6
+ {}^6C_2\,x^4\,(x^2 - 1)
+ {}^6C_4\,x^2\,(x^2 - 1)^2
+ {}^6C_6\,(x^2 - 1)^3
\Bigr].
$$
Step 5: Evaluate the Binomial Coefficients
Recall ${}^6C_0 = 1,\; {}^6C_2 = 15,\; {}^6C_4 = 15,\; {}^6C_6 = 1.$ Hence:
$$
2 \Bigl[
1\cdot x^6
+ 15\,x^4\,(x^2 - 1)
+ 15\,x^2\,(x^2 - 1)^2
+ 1\cdot (x^2 - 1)^3
\Bigr].
$$
Step 6: Expand and Simplify
Expand each term:
$x^4\,(x^2 - 1) = x^6 - x^4.$
$x^2\,(x^2 - 1)^2 = x^2\,(x^4 - 2x^2 + 1) = x^6 - 2x^4 + x^2.$
$(x^2 - 1)^3 = (x^2 - 1)(x^2 - 1)^2 = (x^2 - 1)(x^4 - 2x^2 + 1)
= x^6 - 3x^4 + 3x^2 -1.$
Combine them carefully:
Inside the brackets:
$$
x^6 + 15(x^6 - x^4)
+ 15(x^6 - 2x^4 + x^2)
+ (x^6 - 3x^4 + 3x^2 - 1).
$$
Collect like terms and then multiply by 2 in the end.
Step 7: Identify the Coefficients of $x^4$ and $x^2$
By careful expansion and combining, one finds:
• Coefficient of $x^4 = \alpha = -96.
• Coefficient of $x^2 = \beta = 36.
Step 8: Compute $\alpha - \beta$
We are asked to find $\alpha - \beta.$ Thus:
$$
\alpha - \beta = -96 - 36 = -132.
$$
Final Answer
$\alpha - \beta = -132$
