Solving for $x^2 - 2$ with $x = \sqrt{2} + 1$
We are given the value of $x$ and asked to find the value of the expression $x^2 - 2$. This involves substitution and algebraic simplification.
Step 1: Substitute the value of $x$
Given $x = \sqrt{2} + 1$. Substitute this into the expression $x^2 - 2$. $x^2 - 2 = (\sqrt{2} + 1)^2 - 2$
Step 2: Expand the squared term
Expand $(\sqrt{2} + 1)^2$ using the formula $(a+b)^2 = a^2 + 2ab + b^2$. Here, $a = \sqrt{2}$ and $b = 1$. $(\sqrt{2} + 1)^2 = (\sqrt{2})^2 + 2(\sqrt{2})(1) + (1)^2$ $(\sqrt{2} + 1)^2 = 2 + 2\sqrt{2} + 1$ $(\sqrt{2} + 1)^2 = 3 + 2\sqrt{2}$
Step 3: Calculate the final expression
Now substitute the expanded value back into the expression from Step 1. $x^2 - 2 = (3 + 2\sqrt{2}) - 2$ $x^2 - 2 = 3 - 2 + 2\sqrt{2}$ $x^2 - 2 = 1 + 2\sqrt{2}$
Conclusion
The value of the expression $x^2 - 2$ when $x = \sqrt{2} + 1$ is $1 + 2\sqrt{2}$.
Similar Questions
- If $x = 2 + \sqrt{3}$, find $x - \frac{1}{x}$.
- Simplify:
$\sqrt{12} + \sqrt{75}$ - What will come in place of ? to satisfy the equation : $(\sqrt{6}-1)^2 = ? - 2\sqrt{6}$
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$\frac{\sqrt{5+2\sqrt{6}}}{\sqrt{5-2\sqrt{6}}}$ - The value of $\sqrt{11 - 2\sqrt{30}} - \frac{1}{\sqrt{11 - 2\sqrt{30}}}$ is:
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Rationalize:
$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}$- Simplify:
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Important Questions from Surds and Indices
The value of \(\frac{{{{\left( {251} \right)}^3} + {{\left( {249} \right)}^3}}}{{25.1 \times 25.1 - 624.99 + 24.9 \times 24.9}}\) is 5 × 10 k , where the value of k is :
Find the value of m in \(\left(\frac{2}{7}\right)^{-3} \times \left(\frac{2}{7}\right)^{-5}=\left (\frac{2}{7}\right)^{-3m+1}\)
If √625 = 25; then√(.00000625/25)is:
A. 0.0025
B. 0.001
C. 0.0001
D. 0.0005Find the value of:
\(\sqrt{150}-\sqrt{54}-\sqrt{24}\)
If \(\sqrt{4624}=68\) , then the value of:
\(\sqrt{46.24}+\sqrt{0.4624}+\sqrt{0.004624}\)
