To find the value of $\alpha^2 + \beta^2$, we can use the identities involving sums and products of roots.

Given:

1. $\alpha + \beta = 4$
2. $\alpha \beta = 10$

The identity for $\alpha^2 + \beta^2$ in terms of $\alpha + \beta$ and $\alpha \beta$ is: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2 \alpha \beta$

Step-by-Step Calculation

1. Substitute $\alpha + \beta = 4$ and $\alpha \beta = 10$ into the formula: $\alpha^2 + \beta^2 = (4)^2 - 2 \cdot 10$
2. Calculate $(4)^2$: $(4)^2 = 16$
3. Calculate $2 \cdot 10$: $2 \cdot 10 = 20$
4. Substitute back: $\alpha^2 + \beta^2 = 16 - 20 = -4$

Conclusion

The value of $\alpha^2 + \beta^2$ is: $\alpha^2 + \beta^2 = -4$

Would you like further clarification or have additional questions?

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Here are five related questions to deepen your understanding:

1. What is the value of $\alpha^3 + \beta^3$ given $\alpha + \beta = 4$ and $\alpha \beta = 10$?
2. How can you find $(\alpha - \beta)^2$ using $\alpha + \beta$ and $\alpha \beta$?
3. If $\alpha$ and $\beta$ are roots of a quadratic equation, what would the equation be?
4. How does the sign of $\alpha \beta$ affect the value of $\alpha^2 + \beta^2$?
5. Can $\alpha$ and $\beta$ be real numbers with the given values of $\alpha + \beta$ and $\alpha \beta$?

Tip: When working with sum and product identities, try to remember key formulas like $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$ for quick calculations.