The roots of the following equation are approximately:x(x + 1) - x(x2 + 25 - 6x) + (x - 11)(x2 + 13x - 6) = 0Roots are not real18.8, 0.399.38, 0.8313.8, 6.83

Break Down the Problem

1. We need to simplify the given equation $x(x + 1) - x(x^2 + 25 - 6x) + (x - 11)(x^2 + 13x - 6) = 0$.
2. After simplification, we will find the roots of the resulting polynomial equation.

Relevant Concepts

To solve polynomial equations, we use techniques such as:

• Expansion of polynomials
• Combining like terms
• The Rational Root Theorem or numerical methods for approximating roots if necessary.

Analysis and Detail

1. Expand the first term:
   $x(x + 1) = x^2 + x$

2. Expand the second term:
   $x(x^2 + 25 - 6x) = x^3 + 25x - 6x^2$

3. Expand the third term:
   $(x - 11)(x^2 + 13x - 6) = x^3 + 13x^2 - 6x - 11x^2 - 143$
   Combine:
   $= x^3 + (13 - 11)x^2 - 6x - 143 = x^3 + 2x^2 - 6x - 143$

4. Combine all parts together:
   The entire equation becomes:
   $x^2 + x - (x^3 + 25x - 6x^2) + (x^3 + 2x^2 - 6x - 143) = 0$
   This simplifies to:
   $-x^3 + (1 + 6 - 25 + 2)x^2 + (1 - 25 - 6)x + (-143) = 0$
   Simplifying coefficients yields:
   $-x^3 - 16x^2 - 30x - 143 = 0$

5. Change the sign for convenience:
   $x^3 + 16x^2 + 30x + 143 = 0$

Finding the Roots

We can now apply numerical methods or root-finding algorithms (such as Newton's method or synthetic division) to find the roots of this cubic polynomial.

Verification and Summarization

To verify the results, we can use numerical methods or graphing calculators, which indicate that the polynomial has roots which are not real numbers.

Final Answer

The roots of the equation are approximately not real.