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If alpha is a root of the equation 2x(2x...

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

A

`3alpha^(3)+4alpha`

B

`4alpha^(3)-3alpha`

C

`-2alpha(alpha+1)`

D

None of these

Text Solution

AI Generated Solution

The correct Answer is:
To find the other root of the equation \(2x(2x+1)=1\) given that \(\alpha\) is one root, we can follow these steps: ### Step 1: Rewrite the equation Start with the equation: \[ 2x(2x + 1) = 1 \] Rearranging gives: \[ 2x(2x + 1) - 1 = 0 \] ### Step 2: Expand the equation Expanding the left-hand side: \[ 4x^2 + 2x - 1 = 0 \] ### Step 3: Identify coefficients Here, we can identify the coefficients: - \(a = 4\) - \(b = 2\) - \(c = -1\) ### Step 4: Use Vieta's formulas According to Vieta's formulas, if \(\alpha\) is one root, then the other root \(\beta\) can be found using: \[ \alpha + \beta = -\frac{b}{a} \] Substituting the values: \[ \alpha + \beta = -\frac{2}{4} = -\frac{1}{2} \] ### Step 5: Solve for the other root Now, we can express \(\beta\) in terms of \(\alpha\): \[ \beta = -\frac{1}{2} - \alpha \] ### Step 6: Conclusion Thus, the other root \(\beta\) is: \[ \beta = -\frac{1}{2} - \alpha \]
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