If k is one of the roots of the equation `x(x+1)+1=0`, then what is its other root?

Options

A. 1
B. `-k`
C. `k^(2)`
D. `-k^(2)`

Answer

To find the other root of the equation \( x(x + 1) + 1 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation in standard quadratic form:
\[ x(x + 1) + 1 = 0 \]
This expands to:
\[ x^2 + x + 1 = 0 \]

### Step 2: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), identify the coefficients:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 1 \)

### Step 3: Calculate the discriminant
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]

### Step 4: Find the roots using the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-1 \pm \sqrt{-3}}{2 \cdot 1} = \frac{-1 \pm \sqrt{3}i}{2} \]

### Step 5: Identify the roots
From the above calculation, the two roots are:
\[ k = \frac{-1 + \sqrt{3}i}{2} \quad \text{and} \quad \beta = \frac{-1 - \sqrt{3}i}{2} \]
Thus, if \( k \) is one of the roots, the other root \( \beta \) is:
\[ \beta = \frac{-1 - \sqrt{3}i}{2} \]

### Final Answer
The other root is:
\[ \frac{-1 - \sqrt{3}i}{2} \]

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