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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.

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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 \( ax^3 + bx^2 + cx + d = 0 \).

Solve the following problems:

1. Problem 48: Solve the equation ax^2 + bx - 2 = 0, given that 8a + 6b = 0. For which values of a and b do we have a unique solution?
2. Problem 49: Consider the polynomial x^3 + 4x^2 - 5x + 8. It has three roots a, b, and c. Without calculating these roots, determine: (a) a + b + c, (b) abc, (c) 1/a + 1/b + 1/c.
3. Problem 50: Consider the polynomial x^3 + 10x + 5. It has three roots a, b, and c. Without calculating these roots, determine: (a) a + b + c, (b) abc, (c) 1/a + 1/b + 1/c.
4. Problem 51: Consider the polynomial 2x^3 - 5x^2 - 3x + 2. It has three roots a, b, and c. Without calculating these roots, determine: (a) a + b + c, (b) abc, (c) 1/a + 1/b + 1/c, (d) (a + b + c)^2, (e) a^2 + b^2 + c^2.

Learn to solve polynomial equations by determining sums and products of roots using Vieta's Formulas. Explore detailed solutions for polynomials such as x^3 + 4x^2 - 5x + 8 and more, without calculating roots. Suitable for students in Grades 10-12.

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