Let's start with the given equation:
[tex]\[ x + \frac{1}{x} = 7 \][/tex]
We need to find the value of:
[tex]\[ x^2 - \frac{1}{x^2} \][/tex]
First, let's square both sides of the given equation:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = 7^2 \][/tex]
Expanding the left-hand side, we get:
[tex]\[ x^2 + 2\cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 49 \][/tex]
Notice that [tex]\( x \cdot \frac{1}{x} = 1 \)[/tex], which simplifies our expression:
[tex]\[ x^2 + 2 + \frac{1}{x^2} = 49 \][/tex]
Subtract 2 from both sides to isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[ x^2 + \frac{1}{x^2} = 49 - 2 \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} = 47 \][/tex]
Now, we need [tex]\( x^2 - \frac{1}{x^2} \)[/tex]. To find this, we use the following identity:
[tex]\[ (x + \frac{1}{x})^2 - (x - \frac{1}{x})^2 = 4 \cdot x \cdot \frac{1}{x} \][/tex]
[tex]\[ (x + \frac{1}{x})^2 - (x - \frac{1}{x})^2 = 4 \][/tex]
We already know that:
[tex]\[ (x + \frac{1}{x})^2 = 49 \][/tex]
Let [tex]\( z = x - \frac{1}{x} \)[/tex]. Then we have:
[tex]\[ (x - \frac{1}{x})^2 \][/tex]
From the identity:
[tex]\[ 49 - (x - \frac{1}{x})^2 = 4 \][/tex]
[tex]\[ (x - \frac{1}{x})^2 = 49 - 4 \][/tex]
[tex]\[ (x - \frac{1}{x})^2 = 45 \][/tex]
Let’s take the square root out:
[tex]\[ x - \frac{1}{x} = \pm \sqrt{45} \][/tex]
We recognize that:
[tex]\[ x^2 - \frac{1}{x^2} = (x - \frac{1}{x})(x + \frac{1}{x}) \][/tex]
So, we calculate:
[tex]\[ x^2 - \frac{1}{x^2} = (\pm \sqrt{45}) (7) \][/tex]
[tex]\[ x^2 - \frac{1}{x^2} = 47 \][/tex]
Thus, the value of [tex]\( x^2 - \frac{1}{x^2} \)[/tex] is:
[tex]\[ \boxed{47} \][/tex]
