The question is asking for the sum of the squares of the roots of the quadratic equation:

$x^2 - 8x + 5 = 0$

We can calculate the sum of the squares of the roots using the following formula:

$\text{Sum of squares of the roots} = (\alpha^2 + \beta^2) = (\alpha + \beta)^2 - 2\alpha\beta$

Where:

• $\alpha$ and $\beta$ are the roots of the equation.
• From Vieta’s relations for a quadratic equation $ax^2 + bx + c = 0$:
  • $\alpha + \beta = -\frac{b}{a}$
  • $\alpha\beta = \frac{c}{a}$

In this case, for the quadratic $x^2 - 8x + 5 = 0$:

• $a = 1$, $b = -8$, and $c = 5$

Now, let's calculate:

1. $\alpha + \beta = -\frac{-8}{1} = 8$
2. $\alpha\beta = \frac{5}{1} = 5$

Using the formula for the sum of squares:

$\alpha^2 + \beta^2 = (8)^2 - 2(5) = 64 - 10 = 54$

Thus, the sum of the squares of the roots is 54.

Would you like a more detailed explanation, or do you have any further questions? Here are some related questions you might find interesting:

1. How do you find the actual roots of a quadratic equation?
2. What is the relationship between the coefficients and roots of a quadratic equation?
3. How can you apply the quadratic formula to solve a quadratic equation?
4. What are the different ways to factor a quadratic equation?
5. What happens if the discriminant of a quadratic equation is negative?

Tip: Always remember Vieta’s formulas when working with roots of quadratic equations!