Math Problem Statement

(\sum_{i=4}^{6}(3-2x_{i}))^{2}-2\sum_{i=1}^{6}6x_{i}

Solution

Let's break down the expression step by step and solve it.

The given expression is:

(i=46(32xi))22i=166xi\left(\sum_{i=4}^{6} (3 - 2x_i)\right)^2 - 2\sum_{i=1}^{6} 6x_i

Step 1: Evaluate the summation i=46(32xi)\sum_{i=4}^{6} (3 - 2x_i)

This summation expands as:

i=46(32xi)=(32x4)+(32x5)+(32x6)\sum_{i=4}^{6} (3 - 2x_i) = (3 - 2x_4) + (3 - 2x_5) + (3 - 2x_6)

So,

i=46(32xi)=92(x4+x5+x6)\sum_{i=4}^{6} (3 - 2x_i) = 9 - 2(x_4 + x_5 + x_6)

Step 2: Square the result

(i=46(32xi))2=(92(x4+x5+x6))2\left(\sum_{i=4}^{6} (3 - 2x_i)\right)^2 = \left(9 - 2(x_4 + x_5 + x_6)\right)^2

Step 3: Evaluate the summation i=166xi\sum_{i=1}^{6} 6x_i

This summation expands as:

i=166xi=6(x1+x2+x3+x4+x5+x6)\sum_{i=1}^{6} 6x_i = 6(x_1 + x_2 + x_3 + x_4 + x_5 + x_6)

Step 4: Multiply the result by 2-2

2i=166xi=12(x1+x2+x3+x4+x5+x6)-2\sum_{i=1}^{6} 6x_i = -12(x_1 + x_2 + x_3 + x_4 + x_5 + x_6)

Step 5: Combine all parts

The full expression now becomes:

(92(x4+x5+x6))212(x1+x2+x3+x4+x5+x6)\left(9 - 2(x_4 + x_5 + x_6)\right)^2 - 12(x_1 + x_2 + x_3 + x_4 + x_5 + x_6)

This is the simplified form of the expression.

Would you like more details or further steps?

Related Questions:

  1. How do you expand and simplify a squared binomial like (ab)2(a-b)^2?
  2. How would this expression change if the summation limits were different?
  3. What if there were additional terms inside the summation? How would it affect the final result?
  4. Can you factorize the expression (92(x4+x5+x6))2\left(9 - 2(x_4 + x_5 + x_6)\right)^2?
  5. How does the expression simplify if xix_i are specific numbers?

Tip:

When working with summations, always ensure to evaluate them carefully before moving on to further operations like squaring or multiplication.

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Math Problem Analysis

Mathematical Concepts

Summation
Binomial Expansion
Quadratic Equation

Formulas

Binomial expansion formula
Quadratic equation formula

Theorems

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Suitable Grade Level

Advanced High School