Let a,b,c be real numbers in G.P. such that a and c are positive , then the roots of the equation ` ax^(2) +bx+c=0`

Options

A. Are real and in the ratio b : ac
B. Are real
C. Are imaginary are in ration ` 1 : omega` is a non-real complex cubic root of constant
D. Are imaginary and are in the ration ` omega^(2) :1 ` with usual notation

Answer

To solve the problem step by step, we will analyze the given information and derive the roots of the quadratic equation \( ax^2 + bx + c = 0 \) where \( a, b, c \) are in geometric progression (G.P.) and \( a \) and \( c \) are positive.

### Step 1: Express \( b \) and \( c \) in terms of \( a \) and the common ratio \( r \)
Since \( a, b, c \) are in G.P., we can express:
- \( b = ar \) (where \( r \) is the common ratio)
- \( c = ar^2 \)

### Step 2: Substitute \( b \) and \( c \) into the quadratic equation
The quadratic equation becomes:
\[ ax^2 + arx + ar^2 = 0 \]
We can factor out \( a \) (since \( a \neq 0 \)):
\[ a(x^2 + rx + r^2) = 0 \]
This simplifies to:
\[ x^2 + rx + r^2 = 0 \]

### Step 3: Use the quadratic formula to find the roots
The quadratic formula states that the roots of \( Ax^2 + Bx + C = 0 \) are given by:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
For our equation \( x^2 + rx + r^2 = 0 \):
- \( A = 1 \)
- \( B = r \)
- \( C = r^2 \)

Substituting these values into the formula gives:
\[ x = \frac{-r \pm \sqrt{r^2 - 4 \cdot 1 \cdot r^2}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{-r \pm \sqrt{r^2 - 4r^2}}{2} \]
\[ x = \frac{-r \pm \sqrt{-3r^2}}{2} \]
\[ x = \frac{-r \pm r\sqrt{-3}}{2} \]
\[ x = \frac{-r \pm r\sqrt{3}i}{2} \]

### Step 4: Factor out \( r \)
Factoring \( r \) out gives:
\[ x = r \left( \frac{-1 \pm \sqrt{3}i}{2} \right) \]

### Step 5: Identify the roots
Let \( \omega = \frac{-1 + \sqrt{3}i}{2} \) and \( \omega^2 = \frac{-1 - \sqrt{3}i}{2} \), which are the complex cube roots of unity. Thus, the roots can be expressed as:
\[ x_1 = r\omega, \quad x_2 = r\omega^2 \]

### Step 6: Find the ratio of the roots
The ratio of the roots \( x_1 \) and \( x_2 \) is:
\[ \frac{x_1}{x_2} = \frac{r\omega}{r\omega^2} = \frac{\omega}{\omega^2} \]
Since \( \omega^3 = 1 \), we have:
\[ \frac{\omega}{\omega^2} = \frac{1}{\omega} \]

### Step 7: Simplify the ratio
Multiplying by \( \frac{\omega^2}{\omega^2} \) gives:
\[ \frac{1}{\omega} = \omega^2 \]
Thus, the ratio of the roots is \( \omega^2 : 1 \).

### Conclusion
The roots of the equation \( ax^2 + bx + c = 0 \) are \( r\omega \) and \( r\omega^2 \) with a ratio of \( \omega^2 : 1 \).

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