To find the value of $\alpha^2 + \beta^2$ given that $\alpha + \beta = 4$ and $\alpha \beta = 10$, we can use the identity:

$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2 \alpha \beta$

Step-by-Step Solution

1. Calculate $(\alpha + \beta)^2$: $(\alpha + \beta)^2 = 4^2 = 16$

2. Calculate $2 \alpha \beta$: $2 \alpha \beta = 2 \times 10 = 20$

3. Substitute into the identity: $\alpha^2 + \beta^2 = 16 - 20 = -4$

Final Answer

$\alpha^2 + \beta^2 = -4$

Would you like further clarification on any steps?

Here are some related questions for additional practice:

1. How would you find $\alpha^3 + \beta^3$ given $\alpha + \beta$ and $\alpha \beta$?
2. What is the sum of the roots and the product of the roots for a quadratic equation?
3. How would you factor a quadratic equation if you know $\alpha + \beta$ and $\alpha \beta$?
4. Can $\alpha$ and $\beta$ be real numbers if $\alpha^2 + \beta^2 = -4$?
5. What is the discriminant of a quadratic equation and how does it relate to the nature of roots?

Tip: Remember that for real numbers, $\alpha^2 + \beta^2$ being negative implies complex roots for the variables $\alpha$ and $\beta$.