Use the equation $x^2 + \frac{1}{x^2} = 5$ to evaluate the expression $(x + \frac{1}{x})^2$.
Understand the Problem
The question asks to use the given equation $x^2 + \frac{1}{x^2} = 5$ to find the value of the expression $(x + \frac{1}{x})^2$. This involves algebraic manipulation to relate the given equation to the expression we want to evaluate.
Answer
$(x + \frac{1}{x})^2 = 7$
Answer for screen readers
$(x + \frac{1}{x})^2 = 7$
Steps to Solve
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Expand the target expression Expand the target expression $(x + \frac{1}{x})^2$ using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In our case, $a = x$ and $b = \frac{1}{x}$. $$ (x + \frac{1}{x})^2 = x^2 + 2(x)(\frac{1}{x}) + (\frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2} $$
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Rearrange the expanded expression Rearrange the terms to group $x^2$ and $\frac{1}{x^2}$ together. $$ (x + \frac{1}{x})^2 = x^2 + \frac{1}{x^2} + 2 $$
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Substitute the given value Substitute the given value $x^2 + \frac{1}{x^2} = 5$ into the expression. $$ (x + \frac{1}{x})^2 = 5 + 2 $$
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Simplify Simplify the expression to get the final answer. $$ (x + \frac{1}{x})^2 = 7 $$
$(x + \frac{1}{x})^2 = 7$
More Information
The problem was solved by expanding the expression and using the provided equation to substitute and find the value.
Tips
A common mistake is not expanding $(x + \frac{1}{x})^2$ correctly and missing the middle term $2(x)(\frac{1}{x}) = 2$. Another mistake is trying to solve for $x$ first. The key to avoiding these mistakes is carefully applying the binomial expansion formula and recognizing that we don't need to find the value of x.
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