If the quadratic equations `ax^(2)+bx+c=0(a,b,c epsilon R, a !=0)` and `x^(2)+4x+5=0` have a common root, then a,b,c must satisfy the relations

Options

A. `agtbgtc`
B. `altbltc`
C. `a=k,b=4k,c=5k(k epsilon R, k!=0)`
D. `b^(2)-4ac` is negative.

Answer

To solve the problem, we need to find the relations that the coefficients \(a\), \(b\), and \(c\) must satisfy given that the quadratic equations \(ax^2 + bx + c = 0\) and \(x^2 + 4x + 5 = 0\) have a common root.

### Step 1: Identify the roots of the second quadratic equation
The second quadratic equation is:
\[ x^2 + 4x + 5 = 0 \]
To find the roots, we calculate the discriminant \(D\):
\[ D = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4 \]
Since the discriminant is negative, the roots are complex. The roots can be expressed as:
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{-4}}{2 \cdot 1} = \frac{-4 \pm 2i}{2} = -2 \pm i \]
Thus, the roots are:
\[ \alpha = -2 + i \quad \text{and} \quad \beta = -2 - i \]

### Step 2: Assume a common root
Let’s assume that the common root of both equations is \(\alpha = -2 + i\). Since the coefficients \(a\), \(b\), and \(c\) are real, the other root of the first equation must be the complex conjugate \(\beta = -2 - i\).

### Step 3: Set up the relations for the coefficients
For the quadratic equation \(ax^2 + bx + c = 0\), if \(\alpha\) is a root, then:
\[ a\alpha^2 + b\alpha + c = 0 \]
Substituting \(\alpha = -2 + i\):
\[ a(-2 + i)^2 + b(-2 + i) + c = 0 \]
Calculating \((-2 + i)^2\):
\[ (-2 + i)^2 = 4 - 4i - 1 = 3 - 4i \]
Thus, substituting back gives:
\[ a(3 - 4i) + b(-2 + i) + c = 0 \]
Separating real and imaginary parts:
\[ (3a - 2b + c) + (-4a + b)i = 0 \]
This leads to two equations:
1. \(3a - 2b + c = 0\) (real part)
2. \(-4a + b = 0\) (imaginary part)

### Step 4: Solve the equations
From the second equation, we can express \(b\) in terms of \(a\):
\[ b = 4a \]
Substituting \(b\) into the first equation:
\[ 3a - 2(4a) + c = 0 \implies 3a - 8a + c = 0 \implies -5a + c = 0 \implies c = 5a \]

### Step 5: Final relations
Thus, we have:
\[ b = 4a \quad \text{and} \quad c = 5a \]
This means that \(a\), \(b\), and \(c\) must satisfy the relations:
\[ a : b : c = 1 : 4 : 5 \]