Deriving the Quadratic Equation from Roots

A quadratic equation can be formed using its roots, $\alpha$ and $\beta$, with the standard formula: $x^2 - (\alpha + \beta)x + \alpha \beta = 0$ Where:

• $(\alpha + \beta)$ is the sum of the roots.
• $\alpha \beta$ is the product of the roots.

In this problem, the given roots are $\alpha = \frac{1}{\sqrt{2}}$ and $\beta = \frac{1}{\sqrt{2}}$.

Calculating Sum and Product of Roots

Sum of Roots: $ \alpha + \beta = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} $ Simplifying, $ \frac{2}{\sqrt{2}} = \frac{2 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2} $. So, the sum of the roots is $ \sqrt{2} $.

Product of Roots: $ \alpha \beta = \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} = \frac{1}{(\sqrt{2})^2} = \frac{1}{2} $. So, the product of the roots is $ \frac{1}{2} $.

Forming the Quadratic Equation

Substitute the sum and product into the standard formula: $ x^2 - (\sqrt{2})x + \frac{1}{2} = 0 $

To eliminate the fraction and match the format of the options, multiply the entire equation by 2: $ 2 \times (x^2 - \sqrt{2}x + \frac{1}{2}) = 2 \times 0 $ $ 2x^2 - 2\sqrt{2}x + 1 = 0 $

Matching with Options

Compare the derived equation, $2x^2 - 2\sqrt{2}x + 1 = 0$, with the given options:

• Option 1: $2x^2 - (2\sqrt{2})x + 1 = 0$ (Matches)
• Option 2: $3x^2 - (2\sqrt{2})x - 1 = 0$ (Does not match)
• Option 3: $2x^2 - (4\sqrt{2})x - 1 = 0$ (Does not match)
• Option 4: $2x^2 + (2\sqrt{5})x + 1 = 0$ (Does not match)

The derived quadratic equation matches Option 1.