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

1. Break Down the Problem

We start with the equation:

$x(x + 1) - x(x^2 + 25 - 6x) + (x - 11)(x^2 + 13x - 6) = 0$

We will simplify this equation step by step, expand the terms, and combine like terms.

2. Relevant Concepts

We will use algebraic manipulation to combine the terms.

3. Analysis and Detail

1. Expand each term:

   • First term: $x(x + 1) = x^2 + x$

   • Second term: $-x(x^2 + 25 - 6x) = -x^3 - 25x + 6x^2$

   • Third term: $(x - 11)(x^2 + 13x - 6) = x^3 + 13x^2 - 6x - 11x^2 - 143x + 66 = x^3 + 2x^2 - 149x + 66$

2. Combine all the expanded terms: $x^2 + x - x^3 - 25x + 6x^2 + x^3 + 2x^2 - 149x + 66 = 0$

   Combining like terms results in: $(0)x^3 + (1 + 6 + 2)x^2 + (1 - 25 - 149)x + 66 = 0$

   Simplifying this gives: $9x^2 - 173x + 66 = 0$

4. Verify and Summarize

This is a quadratic equation in standard form $ax^2 + bx + c = 0$, where $a = 9$, $b = -173$, and $c = 66$.

We can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

5. Calculate the Roots

1. Calculate the discriminant: $D = b^2 - 4ac = (-173)^2 - 4 \times 9 \times 66$ $D = 29929 - 2376 = 27553$

2. Now calculate the roots: $x = \frac{173 \pm \sqrt{27553}}{18}$

   • Calculate $\sqrt{27553}$: $\sqrt{27553} \approx 165.78$

   • Therefore: $x = \frac{173 \pm 165.78}{18}$

   Calculating both possible values:

   • First root: $x_1 = \frac{173 + 165.78}{18} = \frac{338.78}{18} \approx 18.8$
   • Second root: $x_2 = \frac{173 - 165.78}{18} = \frac{7.22}{18} \approx 0.40$

Final Answer

The roots of the equation are approximately:

• $x \approx 18.8$
• $x \approx 0.40$

Thus, the roots are not both real and the final answer includes the values.