Let `a,b,c,d,e,f,g,h` be distinct elements in the set `{-7,-5,-3,-2,2,4,6,13}`. The minimum value of `(a+b+c+d)^2 + (e+ f + g + h)^2` is:(1) 30 (2) 32 ( 3) 34 ( 4) 40

Options

A. 30
B. 32
C. 34
D. 40

Explanation

To solve the problem, we need to minimize the expression \((a+b+c+d)^2 + (e+f+g+h)^2\) given that \(a, b, c, d, e, f, g, h\) are distinct elements from the set \(\{-7, -5, -3, -2, 2, 4, 6, 13\}\).

### Step 1: Define the variables
Let:
- \(x = a + b + c + d\)
- \(y = e + f + g + h\)

Since \(a, b, c, d, e, f, g, h\) are all distinct elements from the set, we have:
\[ x + y = -7 - 5 - 3 - 2 + 2 + 4 + 6 + 13 = 8 \]

### Step 2: Express \(y\) in terms of \(x\)
From the equation \(x + y = 8\), we can express \(y\) as:
\[ y = 8 - x \]

### Step 3: Substitute \(y\) into the expression
Now, we substitute \(y\) into the expression we want to minimize:
\[ (x)^2 + (8 - x)^2 \]

### Step 4: Expand the expression
Expanding the expression gives:
\[ x^2 + (8 - x)^2 = x^2 + (64 - 16x + x^2) = 2x^2 - 16x + 64 \]

### Step 5: Simplify the expression
We can simplify this to:
\[ 2x^2 - 16x + 64 \]

### Step 6: Find the minimum value
To find the minimum value of the quadratic expression \(2x^2 - 16x + 64\), we can complete the square or use the vertex formula. The vertex \(x\) of a parabola given by \(ax^2 + bx + c\) occurs at:
\[ x = -\frac{b}{2a} = -\frac{-16}{2 \cdot 2} = \frac{16}{4} = 4 \]

### Step 7: Calculate the minimum value
Now we substitute \(x = 4\) back into the expression:
\[ 2(4)^2 - 16(4) + 64 = 2(16) - 64 + 64 = 32 \]

### Conclusion
Thus, the minimum value of \((a+b+c+d)^2 + (e+f+g+h)^2\) is \(32\).

### Final Answer
The minimum value is \(\boxed{32}\).

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