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The problem you've uploaded presents a quadratic equation:

\[
x^2 + ax + b = 0
\]

You are given that the equation has two roots, \(x_1\) and \(x_2\), and that the following conditions hold:

1. The equation \( x_1 x_2 - (x_1 + x_2) = 6 \).
2. \( x_1 = x_2^2 + 2 \).

Your task is to determine the values of \(a\) and \(b\).

### Steps to Solve:
1. Use Vieta's formulas, which relate the coefficients of the quadratic equation to the sums and products of its roots.
   - \( x_1 + x_2 = -a \) (sum of roots)
   - \( x_1 x_2 = b \) (product of roots)
   
2. Substitute these into the given condition \( x_1 x_2 - (x_1 + x_2) = 6 \).
   
3. Use the second condition \( x_1 = x_2^2 + 2 \) to substitute for \(x_1\) and solve the resulting system.

Let's go ahead and solve this problem. sympy import symbols, Eq, solve

# Define variables
x1, x2, a, b = symbols('x1 x2 a b')

# Equations based on the problem
eq1 = Eq(x1 + x2, -a)  # sum of roots
eq2 = Eq(x1 * x2, b)  # product of roots
eq3 = Eq(x1 * x2 - (x1 + x2), 6)  # given condition x1 x2 - (x1 + x2) = 6
eq4 = Eq(x1, x2**2 + 2)  # given condition x1 = x2^2 + 2

# Solve the system of equations
solutions = solve([eq1, eq2, eq3, eq4], (x1, x2, a, b))
solutionsThe solution involves complex expressions that include roots and cube roots, making it challenging to provide a simple closed form for \(a\) and \(b\) directly. However, based on the equations and conditions provided, this suggests that the values of \(a\) and \(b\) are interconnected through these intricate formulas.

Would you like me to continue simplifying the expression, or would you like further assistance with a specific part of the solution? Let me know if you'd like a detailed breakdown of the steps!

Cho phương trình bậc hai x^2 + ax + b = 0. Biết rằng phương trình có 2 nghiệm x_1, x_2 thoả mãn x_1 x_2 - (x_1 + x_2) = 6 và x_1 = x_2^2 + 2. Tìm a và b.

This problem involves solving a quadratic equation x^2 + ax + b = 0 with specific conditions related to its roots. Learn how to apply Vieta's formulas and solve for the coefficients a and b based on the given conditions. Suitable for students in Grades 9-12.

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