If `alpha` is a root of the equation `2x(2x+1)=1`, then the other root is

Options

A. `3alpha^(3)+4alpha`
B. `4alpha^(3)-3alpha`
C. `-2alpha(alpha+1)`
D. None of these

Answer

To find the other root of the equation \(2x(2x+1) = 1\) given that \(\alpha\) is one of the roots, we can follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation in standard polynomial form:
\[ 2x(2x + 1) - 1 = 0 \]
Expanding this gives:
\[ 4x^2 + 2x - 1 = 0 \]

### Step 2: Identify coefficients
From the standard form \(ax^2 + bx + c = 0\), we can identify:
- \(a = 4\)
- \(b = 2\)
- \(c = -1\)

### Step 3: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation \(ax^2 + bx + c = 0\):
- The sum of the roots \((\alpha + \beta)\) is given by \(-\frac{b}{a}\).
- The product of the roots \((\alpha \cdot \beta)\) is given by \(\frac{c}{a}\).

Using the values we identified:
\[ \alpha + \beta = -\frac{2}{4} = -\frac{1}{2} \]

### Step 4: Solve for the other root
We can express \(\beta\) in terms of \(\alpha\):
\[ \beta = -\frac{1}{2} - \alpha \]

### Step 5: Conclusion
Thus, the other root \(\beta\) is:
\[ \beta = -\frac{1}{2} - \alpha \]