Which quadratic equation has solutions $x = 2 + \frac{1}{\sqrt{2}}$ and $x = 2 - \frac{1}{\sqrt{2}}$?

Steps to Solve

1. Write the solutions We are given the solutions $x = 2 + \frac{1}{\sqrt{2}}$ and $x = 2 - \frac{1}{\sqrt{2}}$.

2. Rewrite the solutions to obtain zero on one side Subtract the terms on the right side to get zero on one side. $x - (2 + \frac{1}{\sqrt{2}}) = 0$ and $x - (2 - \frac{1}{\sqrt{2}}) = 0$.

3. Multiply the two factors Multiply the two factors together to create a quadratic equation that has the given roots. $(x - (2 + \frac{1}{\sqrt{2}}))(x - (2 - \frac{1}{\sqrt{2}})) = 0$ Expand the expression: $x^2 - x(2 - \frac{1}{\sqrt{2}}) - x(2 + \frac{1}{\sqrt{2}}) + (2 + \frac{1}{\sqrt{2}})(2 - \frac{1}{\sqrt{2}}) = 0$

4. Simplify the equation Combine like terms: $x^2 - 2x + \frac{x}{\sqrt{2}} - 2x - \frac{x}{\sqrt{2}} + (4 - \frac{2}{\sqrt{2}} + \frac{2}{\sqrt{2}} - \frac{1}{2}) = 0$ $x^2 - 4x + 4 - \frac{1}{2} = 0$ $x^2 - 4x + \frac{8}{2} - \frac{1}{2} = 0$ $x^2 - 4x + \frac{7}{2} = 0$

5. Multiply to remove the fraction Multiply the entire equation by $2$ to eliminate the fraction: $2(x^2 - 4x + \frac{7}{2}) = 2(0)$ $2x^2 - 8x + 7 = 0$