If `alpha,beta` are the roots of equation `2x(2x+1)=1` then `beta=`

Options

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

Answer

To solve the equation \(2x(2x + 1) = 1\) and find the value of \(\beta\) in terms of \(\alpha\), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 2x(2x + 1) = 1 \]
Expanding the left side gives:
\[ 4x^2 + 2x = 1 \]

### Step 2: Rearrange to standard form
Rearranging the equation to bring all terms to one side yields:
\[ 4x^2 + 2x - 1 = 0 \]

### Step 3: Identify coefficients
In the standard quadratic form \(Ax^2 + Bx + C = 0\), we identify:
- \(A = 4\)
- \(B = 2\)
- \(C = -1\)

### Step 4: Use Vieta's formulas
According to Vieta's formulas, for the roots \(\alpha\) and \(\beta\):
- The sum of the roots is given by:
\[ \alpha + \beta = -\frac{B}{A} = -\frac{2}{4} = -\frac{1}{2} \]
- The product of the roots is given by:
\[ \alpha \beta = \frac{C}{A} = \frac{-1}{4} \]

### Step 5: Express \(\beta\) in terms of \(\alpha\)
From the sum of the roots, we can express \(\beta\) in terms of \(\alpha\):
\[ \beta = -\frac{1}{2} - \alpha \]

### Step 6: Substitute \(\alpha\) into the product equation
Now, we can substitute \(\beta\) into the product equation:
\[ \alpha \left(-\frac{1}{2} - \alpha\right) = -\frac{1}{4} \]
Expanding this gives:
\[ -\frac{1}{2}\alpha - \alpha^2 = -\frac{1}{4} \]
Rearranging leads to:
\[ \alpha^2 + \frac{1}{2}\alpha - \frac{1}{4} = 0 \]

### Step 7: Solve for \(\alpha\)
To find \(\alpha\), we can use the quadratic formula:
\[ \alpha = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Here, \(A = 1\), \(B = \frac{1}{2}\), and \(C = -\frac{1}{4}\):
\[ \alpha = \frac{-\frac{1}{2} \pm \sqrt{\left(\frac{1}{2}\right)^2 - 4 \cdot 1 \cdot -\frac{1}{4}}}{2 \cdot 1} \]
Calculating the discriminant:
\[ \left(\frac{1}{2}\right)^2 + 1 = \frac{1}{4} + 1 = \frac{5}{4} \]
Thus,
\[ \alpha = \frac{-\frac{1}{2} \pm \frac{\sqrt{5}}{2}}{2} = \frac{-1 \pm \sqrt{5}}{4} \]

### Step 8: Find \(\beta\)
Using the expression for \(\beta\):
\[ \beta = -\frac{1}{2} - \left(\frac{-1 \pm \sqrt{5}}{4}\right) \]
Calculating this gives:
\[ \beta = -\frac{2}{4} + \frac{1 \mp \sqrt{5}}{4} = \frac{-2 + 1 \mp \sqrt{5}}{4} = \frac{-1 \mp \sqrt{5}}{4} \]

### Final Answer
Thus, \(\beta\) can be expressed in terms of \(\alpha\) as:
\[ \beta = -\frac{1}{2} - \alpha \]