The roots of the equation `x^(3) - 14x^(2) + 56x- 64 = 0 ` are in …… progression

Options

A. `A.G.P`
B. H.P
C. A.P
D. G.P

Answer

To determine the progression of the roots of the equation \( x^3 - 14x^2 + 56x - 64 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients of the polynomial
The given polynomial is \( x^3 - 14x^2 + 56x - 64 = 0 \). Here, we can identify:
- \( a = 1 \) (coefficient of \( x^3 \))
- \( b = -14 \) (coefficient of \( x^2 \))
- \( c = 56 \) (coefficient of \( x \))
- \( d = -64 \) (constant term)

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a cubic equation \( ax^3 + bx^2 + cx + d = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -\frac{b}{a} = 14 \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \alpha\gamma = \frac{c}{a} = 56 \)
- The product of the roots \( \alpha\beta\gamma = -\frac{d}{a} = 64 \)

### Step 3: Assume the roots are in Geometric Progression (GP)
Let the roots be \( \alpha, \beta, \gamma \) such that \( \alpha = a/r, \beta = a, \gamma = ar \) for some \( a \) and \( r \).

### Step 4: Express the sum of the roots
From the sum of the roots:
\[ \alpha + \beta + \gamma = \frac{a}{r} + a + ar = 14 \]
Multiplying through by \( r \):
\[ a + ar + a r^2 = 14r \]
This simplifies to:
\[ a(1 + r + r^2) = 14r \quad \text{(Equation 1)} \]

### Step 5: Express the sum of the products of the roots taken two at a time
From the sum of the products of the roots:
\[ \alpha\beta + \beta\gamma + \alpha\gamma = \left(\frac{a}{r}\right)a + a(ar) + \left(\frac{a}{r}\right)(ar) = \frac{a^2}{r} + a^2 + \frac{a^2}{r} = a^2\left(\frac{1}{r} + 1 + r\right) = 56 \]
This simplifies to:
\[ a^2\left(\frac{1 + r + r^2}{r}\right) = 56 \quad \text{(Equation 2)} \]

### Step 6: Express the product of the roots
From the product of the roots:
\[ \alpha\beta\gamma = \left(\frac{a}{r}\right)a(ar) = a^3 = 64 \]
Thus, we have:
\[ a^3 = 64 \implies a = 4 \]

### Step 7: Substitute \( a \) back into the equations
Substituting \( a = 4 \) into Equation 1:
\[ 4(1 + r + r^2) = 14r \implies 1 + r + r^2 = \frac{14r}{4} = 3.5r \]
Rearranging gives:
\[ r^2 - 2.5r + 1 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula:
\[ r = \frac{-(-2.5) \pm \sqrt{(-2.5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{2.5 \pm \sqrt{6.25 - 4}}{2} = \frac{2.5 \pm \sqrt{2.25}}{2} = \frac{2.5 \pm 1.5}{2} \]
Calculating gives:
\[ r = 2 \quad \text{or} \quad r = 0.5 \]

### Step 9: Determine the roots
If \( r = 2 \):
- The roots are \( \alpha = 2, \beta = 4, \gamma = 8 \) (in GP).

If \( r = 0.5 \):
- The roots are \( \alpha = 8, \beta = 4, \gamma = 2 \) (also in GP).

### Conclusion
Thus, the roots of the equation are in Geometric Progression (GP).