Key Concepts and Formulas

• Standard Form of a Quadratic Equation: A quadratic equation is generally expressed as $ax^2 + bx + c = 0$, where $a \neq 0$.
• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$:
  • $\alpha + \beta = -\frac{b}{a}$ (Sum of roots)
  • $\alpha\beta = \frac{c}{a}$ (Product of roots)
• Root Property: If $\alpha$ is a root of the equation $f(x) = 0$, then $f(\alpha) = 0$.

Step-by-Step Solution

Step 1: Convert the given equation to standard form. The given equation is $2x(2x + 1) = 1$. To apply Vieta's formulas, we must rewrite it in the standard quadratic form $ax^2 + bx + c = 0$. Expanding the left side, we get: $4x^2 + 2x = 1$ Subtracting 1 from both sides gives the standard form: $4x^2 + 2x - 1 = 0$ Here, we can identify the coefficients: $a = 4$, $b = 2$, and $c = -1$.

Step 2: Apply Vieta's formulas to find the sum of the roots. Since $\alpha$ and $\beta$ are the roots of $4x^2 + 2x - 1 = 0$, we can use Vieta's formulas. The sum of the roots is: $\alpha + \beta = -\frac{b}{a}$ Substituting the values of $a$ and $b$, we get: $\alpha + \beta = -\frac{2}{4}$ $\alpha + \beta = -\frac{1}{2}$ Multiplying both sides by 2, we have: $2\alpha + 2\beta = -1 \quad (*)$

Step 3: Utilize the property that $\alpha$ is a root of the equation. Since $\alpha$ is a root of the equation $4x^2 + 2x - 1 = 0$, substituting $x = \alpha$ into the equation must satisfy it: $4\alpha^2 + 2\alpha - 1 = 0 \quad (**)$

Step 4: Substitute and solve for $\beta$ in terms of $\alpha$. We want to express $\beta$ in terms of $\alpha$. We have two equations:

1. $2\alpha + 2\beta = -1$ (from $(*)$)
2. $4\alpha^2 + 2\alpha - 1 = 0$ (from $(**)$) From equation (1), we have $-1 = 2\alpha + 2\beta$. Substituting this into equation (2), we get: $4\alpha^2 + 2\alpha + (2\alpha + 2\beta) = 0$ Combining like terms: $4\alpha^2 + 4\alpha + 2\beta = 0$ Isolating the term with $\beta$: $2\beta = -4\alpha^2 - 4\alpha$ Dividing by 2: $\beta = -2\alpha^2 - 2\alpha$ Factoring out $-2\alpha$: $\beta = -2\alpha(\alpha + 1)$

Thus, the value of $\beta$ in terms of $\alpha$ is $-2\alpha(\alpha + 1)$.

Common Mistakes & Tips

• Standard Form First: Always convert the quadratic equation to the standard form $ax^2 + bx + c = 0$ before applying Vieta's formulas.
• Sign Conventions: Be careful with signs, especially when using Vieta's formulas. Remember that the sum of the roots is $-\frac{b}{a}$.
• Root Substitution: Remember that substituting a root into its equation will always satisfy the equation.

Summary

By using Vieta's formulas and the property that a root satisfies its equation, we can express one root in terms of the other without explicitly solving for the roots. We rewrote the given equation in standard form, applied Vieta's formulas to find a relationship between the sum of the roots, and then used the root property to substitute and solve for $\beta$ in terms of $\alpha$. The final result is $\beta = -2\alpha(\alpha + 1)$.

The final answer is \boxed{-2\alpha(\alpha + 1)}, which corresponds to option (A).