If 2a2 + 2b2 + 2c2 + 4ab + 4bc + 4ca + 3a + 3b + 3c = O and a + b + c -3/2, a b c O, then what is the value of (a + b + c + abc)/(a3 + b3 + c3)?


The given equation is:

\[
2a^2 + 2b^2 + 2c^2 + 4ab + 4bc + 4ca + 3a + 3b + 3c = O
\]

We also have the condition:

\[
a + b + c = \frac{3}{2}
\]

To simplify the expression, notice that the left-hand side of the equation can be rewritten as:

\[
2(a + b + c)^2 + 3(a + b + c) = O
\]

Let's break it down step by step:

1. \( a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2 \), so the first part becomes:
   \[
   2(a + b + c)^2
   \]

2. Then the remaining terms \( 3a + 3b + 3c \) can be factored as:
   \[
   3(a + b + c)
   \]

Thus, the equation becomes:

\[
2(a + b + c)^2 + 3(a + b + c) = O
\]

Substitute \( a + b + c = \frac{3}{2} \) into the equation:

\[
2\left(\frac{3}{2}\right)^2 + 3\left(\frac{3}{2}\right) = O
\]

First, calculate each part:
- \( \left(\frac{3}{2}\right)^2 = \frac{9}{4} \)
- \( 2 \times \frac{9}{4} = \frac{18}{4} = 4.5 \)
- \( 3 \times \frac{3}{2} = 4.5 \)

Now the equation becomes:

\[
4.5 + 4.5 = O
\]

Thus, \( O = 9 \).

Now, we need to find the value of \( \frac{a + b + c + abc}{a^3 + b^3 + c^3} \).

Since \( a + b + c = \frac{3}{2} \) and the equation involving the sum of squares is simplified, you can assume that \( a = b = c \). Hence, \( a = b = c = \frac{1}{2} \).

Substitute these values into \( \frac{a + b + c + abc}{a^3 + b^3 + c^3} \):

1. \( a + b + c = \frac{3}{2} \)
2. \( abc = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \)
3. \( a^3 + b^3 + c^3 = 3 \times \left(\frac{1}{2}\right)^3 = 3 \times \frac{1}{8} = \frac{3}{8} \)

Now compute the value:

\[
\frac{\frac{3}{2} + \frac{1}{8}}{\frac{3}{8}} = \frac{\frac{12}{8} + \frac{1}{8}}{\frac{3}{8}} = \frac{\frac{13}{8}}{\frac{3}{8}} = \frac{13}{3}
\]

Thus, the value of \( \frac{a + b + c + abc}{a^3 + b^3 + c^3} \) is \( \frac{13}{3} \).

Would you like further explanation or have any questions?

Here are some related questions you might explore:
1. How would the solution change if \( a + b + c \) had a different value?
2. What is the geometric interpretation of this equation?
3. Can this be solved using matrices or determinants?
4. How would the expression for \( a^3 + b^3 + c^3 \) change for different values of \( a, b, c \)?
5. What happens if \( a, b, c \) are complex numbers?

**Tip**: Symmetry in equations often suggests that the variables can be assumed equal, simplifying the problem.

If 2a^2 + 2b^2 + 2c^2 + 4ab + 4bc + 4ca + 3a + 3b + 3c = O and a + b + c = 3/2, a, b, c, O, then what is the value of (a + b + c + abc)/(a^3 + b^3 + c^3)?

Learn how to solve the algebraic expression for (a + b + c + abc)/(a^3 + b^3 + c^3) when a + b + c = 3/2 and simplify using symmetry in equations. This problem covers concepts in algebra and quadratic expressions. Suitable for students in Grades 10-12.

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