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16. If [tex]x + \frac{1}{x} = 6[/tex], what is the value of [tex]x^2 + \frac{1}{x^2}[/tex]?

A. 43
B. 34
C. 36
D. 38

Sagot :

GinnyAnswer
To solve for the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex] given that [tex]\( x + \frac{1}{x} = 6 \)[/tex], we will follow a step-by-step approach:

1. Start with the given equation:
[tex]\[ x + \frac{1}{x} = 6 \][/tex]

2. Square both sides of the equation:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = 6^2 \][/tex]

3. Expand the left-hand side using the algebraic identity [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]:
[tex]\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 36 \][/tex]

4. Simplify the expanded equation:
[tex]\[ x^2 + 2 + \frac{1}{x^2} = 36 \][/tex]

5. Isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[ x^2 + \frac{1}{x^2} = 36 - 2 \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} = 34 \][/tex]

Therefore, the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex] is [tex]\( 34 \)[/tex].

The correct answer is:
b) 34
Hi1315

Answer:

34

Step-by-step explanation:

To find  [tex]x^2 + \frac{1}{x^2}[/tex], we start with the given equation [tex]x + \frac{1}{x} = 6 .[/tex]

First, square both sides of the equation:

[tex]\left( x + \frac{1}{x} \right)^2 = 6^2[/tex]

[tex]x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 36[/tex]

[tex]x^2 + 2 + \frac{1}{x^2} = 36[/tex]

Now, subtract 2 from both sides:

[tex]x^2 + \frac{1}{x^2} = 36 - 2[/tex]

[tex]x^2 + \frac{1}{x^2} = 34[/tex]

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