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If [tex]\( x - \frac{1}{x} = 3 \)[/tex], show that [tex]\( x^2 + \frac{1}{x^2} = 11 \)[/tex].

Sagot :

GinnyAnswer
Sure! Let's go through the problem step-by-step to show that [tex]\( x^2 + \frac{1}{x^2} = 11 \)[/tex] given that [tex]\( x - \frac{1}{x} = 3 \)[/tex].

1. Start with the given equation:

[tex]\[ x - \frac{1}{x} = 3 \][/tex]

2. Square both sides of the equation to isolate [tex]\( x^2 \)[/tex] and [tex]\( \frac{1}{x^2} \)[/tex].

[tex]\[ \left( x - \frac{1}{x} \right)^2 = 3^2 \][/tex]

3. Simplify both sides:

The left-hand side is the square of a binomial:

[tex]\[ \left( x - \frac{1}{x} \right)^2 = x^2 - 2 \left( x \cdot \frac{1}{x} \right) + \left( \frac{1}{x} \right)^2 \][/tex]

Using [tex]\( \left( x \cdot \frac{1}{x} = 1 \)[/tex] \), this becomes:

[tex]\[ x^2 - 2 \cdot 1 + \frac{1}{x^2} = x^2 - 2 + \frac{1}{x^2} \][/tex]

The right-hand side simplifies to:

[tex]\[ 3^2 = 9 \][/tex]

4. Substitute back into the equation:

[tex]\[ x^2 - 2 + \frac{1}{x^2} = 9 \][/tex]

5. Add 2 to both sides of the equation to solve for [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:

[tex]\[ x^2 + \frac{1}{x^2} - 2 + 2 = 9 + 2 \][/tex]

Simplifying gives us:

[tex]\[ x^2 + \frac{1}{x^2} = 11 \][/tex]

Thus, we have shown that:

[tex]\[ x^2 + \frac{1}{x^2} = 11 \][/tex]

This completes the proof.
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