If `a+b+c=24`, `a^(2)+b^(2)+c^(2)=210`, `abc=440`. Then the least value of `a-b-c` is

Options

A. `-2`
B. `2`
C. `8`
D. `-14`

Explanation

To solve the problem step by step, we need to use the given equations and properties of symmetric sums.

### Given:
1. \( a + b + c = 24 \) (Equation 1)
2. \( a^2 + b^2 + c^2 = 210 \) (Equation 2)
3. \( abc = 440 \) (Equation 3)

### Step 1: Find \( ab + ac + bc \)

We can use the identity:
\[ a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + ac + bc) \]
Substituting the values from Equation 1 and Equation 2:
\[ 210 = 24^2 - 2(ab + ac + bc) \]
Calculating \( 24^2 \):
\[ 24^2 = 576 \]
So, we have:
\[ 210 = 576 - 2(ab + ac + bc) \]
Rearranging gives:
\[ 2(ab + ac + bc) = 576 - 210 \]
\[ 2(ab + ac + bc) = 366 \]
\[ ab + ac + bc = 183 \]

### Step 2: Form the cubic equation

Using the values we have, we can form the cubic equation whose roots are \( a, b, c \):
\[ x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc = 0 \]
Substituting the known values:
\[ x^3 - 24x^2 + 183x - 440 = 0 \]

### Step 3: Find the roots of the cubic equation

To find the roots, we can use the Rational Root Theorem or synthetic division. Testing possible rational roots, we find:
- Testing \( x = 5 \):
\[ 5^3 - 24(5^2) + 183(5) - 440 = 125 - 600 + 915 - 440 = 0 \]
Thus, \( x = 5 \) is a root.

Now we can factor the cubic equation by dividing it by \( (x - 5) \):
Using synthetic division:
\[ \begin{array}{r|rrrr} 5 & 1 & -24 & 183 & -440 \\ & & 5 & -95 & 440 \\ \hline & 1 & -19 & 88 & 0 \\ \end{array} \]
This gives us:
\[ x^2 - 19x + 88 = 0 \]

### Step 4: Solve the quadratic equation

Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -19, c = 88 \):
\[ x = \frac{19 \pm \sqrt{(-19)^2 - 4 \cdot 1 \cdot 88}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{19 \pm \sqrt{361 - 352}}{2} \]
\[ x = \frac{19 \pm \sqrt{9}}{2} \]
\[ x = \frac{19 \pm 3}{2} \]
This gives us:
\[ x = \frac{22}{2} = 11 \quad \text{and} \quad x = \frac{16}{2} = 8 \]

### Step 5: Roots of the cubic equation

Thus, the roots are \( a = 5, b = 8, c = 11 \) (in any order).

### Step 6: Find the least value of \( a - b - c \)

Calculating:
\[ a - b - c = 5 - 8 - 11 = 5 - 19 = -14 \]

### Final Answer
The least value of \( a - b - c \) is \( \boxed{-14} \).