Here is a breakdown of the math problems visible in the image:

Problem 48

Solve the equation $ax^2 + bx - 2 = 0$, knowing that 8a + 6b = 0. For which values of $a$ and $b$ do we have a unique solution?

Problem 49

Consider the polynomial $x^3 + 4x^2 - 5x + 8$. It has three roots $a$, $b$, and $c$. Without calculating these roots, determine:

1. $a + b + c$
2. $abc$
3. $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Problem 50

Consider the polynomial $x^3 + 10x + 5$. It has three roots $a$, $b$, and $c$. Without calculating these roots, determine:

1. $a + b + c$
2. $abc$
3. $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Problem 51

Consider the polynomial $2x^3 - 5x^2 - 3x + 2$. It has three roots $a$, $b$, and $c$. Without calculating these roots, determine:

1. $a + b + c$
2. $abc$
3. $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$
4. $(a + b + c)^2$
5. $a^2 + b^2 + c^2$

Would you like a detailed solution for any of these problems? Let me know if you have questions or if you want to focus on a specific exercise.

Here are 5 related questions that could expand on these exercises:

1. How can we determine the sum and product of roots for any polynomial of degree 3?
2. What is the relationship between the coefficients of a polynomial and its roots in general terms?
3. How can Vieta’s formulas be applied to polynomials of higher degrees?
4. How can we find the value of $a^2 + b^2 + c^2$ given $a + b + c$ and $abc$?
5. How can we determine whether a polynomial has real or complex roots without solving it?

Tip: For cubic polynomials, Vieta's formulas give the sum of the roots as $-\frac{b}{a}$ and the product of the roots as $-\frac{d}{a}$, assuming the polynomial is $ax^3 + bx^2 + cx + d = 0$.