IDNLearn.com provides a collaborative environment for finding and sharing knowledge. Find in-depth and trustworthy answers to all your questions from our experienced community members.
Sagot :
Let's solve the given problem step-by-step.
We start with the equation:
[tex]\[ x - \frac{1}{x} = 8. \][/tex]
First, we square both sides of the equation to simplify it further:
[tex]\[ \left( x - \frac{1}{x} \right)^2 = 8^2. \][/tex]
Expanding the left-hand side using the algebraic identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex], we get:
[tex]\[ x^2 - 2 \left( x \cdot \frac{1}{x} \right) + \frac{1}{x^2} = 64. \][/tex]
Simplifying the expression:
[tex]\[ x^2 - 2 + \frac{1}{x^2} = 64. \][/tex]
Next, we isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex] by adding 2 to both sides:
[tex]\[ x^2 + \frac{1}{x^2} = 64 + 2. \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} = 66. \][/tex]
Now, we need to find the value of [tex]\( x^2 + \frac{1}{x^2} - 8 \)[/tex]. We substitute the value we just found:
[tex]\[ x^2 + \frac{1}{x^2} - 8 = 66 - 8. \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} - 8 = 58. \][/tex]
Therefore, the correct answer is:
(ii) 58
We start with the equation:
[tex]\[ x - \frac{1}{x} = 8. \][/tex]
First, we square both sides of the equation to simplify it further:
[tex]\[ \left( x - \frac{1}{x} \right)^2 = 8^2. \][/tex]
Expanding the left-hand side using the algebraic identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex], we get:
[tex]\[ x^2 - 2 \left( x \cdot \frac{1}{x} \right) + \frac{1}{x^2} = 64. \][/tex]
Simplifying the expression:
[tex]\[ x^2 - 2 + \frac{1}{x^2} = 64. \][/tex]
Next, we isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex] by adding 2 to both sides:
[tex]\[ x^2 + \frac{1}{x^2} = 64 + 2. \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} = 66. \][/tex]
Now, we need to find the value of [tex]\( x^2 + \frac{1}{x^2} - 8 \)[/tex]. We substitute the value we just found:
[tex]\[ x^2 + \frac{1}{x^2} - 8 = 66 - 8. \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} - 8 = 58. \][/tex]
Therefore, the correct answer is:
(ii) 58
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.