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The problem in the image is:

\[
\text{If } \alpha, \beta \text{ are the roots of } 2x^2 - 2x + 1 = 0, \text{ what is the value of } \alpha + \beta?
\]

The quadratic equation is \(2x^2 - 2x + 1 = 0\). To find \(\alpha + \beta\), the sum of the roots, we can use Vieta's formulas. For a quadratic equation of the form \(ax^2 + bx + c = 0\), the sum of the roots \(\alpha + \beta\) is given by:

\[
\alpha + \beta = -\frac{b}{a}
\]

Here, \(a = 2\), \(b = -2\), and \(c = 1\). Plugging in the values:

\[
\alpha + \beta = -\frac{-2}{2} = \frac{2}{2} = 1
\]

So, the value of \(\alpha + \beta\) is \( \boxed{1} \).

Would you like more details or have any other questions? Here are some related questions for further practice:

1. How do you find the product of the roots \(\alpha \beta\) of the quadratic equation \(2x^2 - 2x + 1 = 0\)?
2. Can you derive the quadratic formula from completing the square?
3. What is the discriminant of \(2x^2 - 2x + 1 = 0\), and what does it tell you about the nature of the roots?
4. How do you solve the quadratic equation \(2x^2 - 2x + 1 = 0\) using the quadratic formula?
5. What is the sum of the roots for the equation \(ax^2 + bx + c = 0\) when \(a = 1\), \(b = 4\), and \(c = 4\)?
6. What happens to the roots of a quadratic equation when the discriminant is negative?
7. Can you graph the function \(f(x) = 2x^2 - 2x + 1\)? Where does it intersect the x-axis?
8. How can Vieta's formulas be applied to cubic and quartic equations?

**Tip:** Always remember that Vieta's formulas are a quick way to find the sum and product of the roots of a quadratic equation without solving the equation directly.

Understand how to find the sum of roots for the quadratic equation 2x^2 - 2x + 1 = 0 using Vieta's formulas. This problem covers key concepts in quadratic equations and Vieta's theorem, providing a detailed step-by-step solution. Suitable for students in Grades 10-12.

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