Rewrite Quadratic Equation

The given equation is $2x(2x + 1) = 1$. First, expand the equation: $4x^2 + 2x = 1$ Now, rearrange it into the standard quadratic form $ax^2 + bx + c = 0$: $4x^2 + 2x - 1 = 0$

Identify Coefficients

From the standard form $4x^2 + 2x - 1 = 0$, we identify the coefficients: $a = 4$ $b = 2$ $c = -1$

Apply Vieta's Formulas

Let the roots of the quadratic equation be $\alpha$ and $\beta$. According to Vieta's formulas, the product of the roots is given by $\frac{c}{a}$. $ \alpha \cdot \beta = \frac{c}{a} $ Substitute the identified coefficients: $ \alpha \cdot \beta = \frac{-1}{4} $

Find the Other Root

We are given that $\alpha$ is one root. We need to find the other root, $\beta$. Using the product of roots relationship: $ \alpha \beta = -\frac{1}{4} $ Since $x=0$ does not satisfy the original equation ($2(0)(2(0)+1) = 0 \neq 1$), $\alpha$ cannot be zero. Therefore, we can divide by $\alpha$: $ \beta = -\frac{1}{4\alpha} $ This result matches option 3.