In quadratic equation `cx^(2) + bx + c = 0` find the ratio `b : c` if the given equation has equal roots

Options

A. `1 : 2`
B. `2 : 1`
C. `4 : 1`
D. `1 : 4`

Answer

To solve the problem of finding the ratio \( b : c \) in the quadratic equation \( cx^2 + bx + c = 0 \) when it has equal roots, we can follow these steps:

### Step 1: Identify the standard form of the quadratic equation
The given quadratic equation is:
\[ cx^2 + bx + c = 0 \]
Here, \( a = c \), \( b = b \), and \( c = c \).

### Step 2: Use the condition for equal roots
For a quadratic equation to have equal roots, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Setting the discriminant equal to zero for equal roots:
\[ b^2 - 4ac = 0 \]

### Step 3: Substitute the values of \( a \), \( b \), and \( c \)
Substituting \( a = c \) into the discriminant equation:
\[ b^2 - 4(c)(c) = 0 \]
This simplifies to:
\[ b^2 - 4c^2 = 0 \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ b^2 = 4c^2 \]

### Step 5: Taking the square root
Taking the square root of both sides:
\[ b = 2c \quad \text{or} \quad b = -2c \]

### Step 6: Finding the ratio
Now, we can express the ratio \( b : c \):
\[ \frac{b}{c} = 2 \quad \text{or} \quad \frac{b}{c} = -2 \]
Thus, the ratio \( b : c \) can be written as:
\[ b : c = 2 : 1 \quad \text{or} \quad b : c = -2 : 1 \]

### Conclusion
The ratio \( b : c \) when the quadratic equation has equal roots is \( 2 : 1 \) (considering positive roots).

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