Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha \beta = \frac{c}{a}$.
• Forming a Quadratic Equation from Roots: If $\alpha'$ and $\beta'$ are the roots of a quadratic equation, the equation can be written as $x^2 - (\alpha' + \beta')x + \alpha' \beta' = 0$.

Step-by-Step Solution

Step 1: Find the coefficients $a$ and $b$ of the given equation

We are given the quadratic equation $ax^2 + bx + 1 = 0$ with roots 2 and 6. We will use Vieta's formulas to find $a$ and $b$.

• Apply Vieta's formula for the sum of roots: The sum of the roots is $2 + 6 = 8$. According to Vieta's formulas, the sum of the roots is $-\frac{b}{a}$. Therefore, we have $8 = -\frac{b}{a}$ $b = -8a \quad (\text{Equation 1})$ Explanation: We use the sum of the roots formula to establish a relationship between $a$ and $b$ based on the given roots.

• Apply Vieta's formula for the product of roots: The product of the roots is $2 \times 6 = 12$. According to Vieta's formulas, the product of the roots is $\frac{1}{a}$. Therefore, we have $12 = \frac{1}{a}$ $a = \frac{1}{12}$ Explanation: We use the product of the roots formula to directly calculate the value of $a$ because the constant term is known.

• Calculate $b$ using the value of $a$: Substitute $a = \frac{1}{12}$ into Equation 1: $b = -8a = -8 \left( \frac{1}{12} \right)$ $b = -\frac{8}{12} = -\frac{2}{3}$ Explanation: Knowing $a$, we can find $b$ from the relationship $b = -8a$.

Summary of coefficients: We have $a = \frac{1}{12}$ and $b = -\frac{2}{3}$.

Step 2: Determine the values of the new roots

The roots of the new quadratic equation are $\frac{1}{2a+b}$ and $\frac{1}{6a+b}$. We will substitute the values of $a$ and $b$ we found in Step 1.

• Calculate the first new root: Substitute $a = \frac{1}{12}$ and $b = -\frac{2}{3}$ into $2a+b$: $2a+b = 2 \left( \frac{1}{12} \right) + \left( -\frac{2}{3} \right)$ $2a+b = \frac{1}{6} - \frac{2}{3}$ $2a+b = \frac{1}{6} - \frac{4}{6} = -\frac{3}{6} = -\frac{1}{2}$ The first new root is $\frac{1}{2a+b} = \frac{1}{-\frac{1}{2}} = -2$ Explanation: We simplify the expression $2a+b$ using fraction arithmetic, then take its reciprocal to find the first root.

• Calculate the second new root: Substitute $a = \frac{1}{12}$ and $b = -\frac{2}{3}$ into $6a+b$: $6a+b = 6 \left( \frac{1}{12} \right) + \left( -\frac{2}{3} \right)$ $6a+b = \frac{1}{2} - \frac{2}{3}$ $6a+b = \frac{3}{6} - \frac{4}{6} = -\frac{1}{6}$ The second new root is $\frac{1}{6a+b} = \frac{1}{-\frac{1}{6}} = -6$ Explanation: Similarly, we simplify $6a+b$ and then take its reciprocal to find the second root.

Summary of new roots: The roots of the new quadratic equation are $-2$ and $-6$.

Step 3: Form the new quadratic equation

We have the roots $\alpha' = -2$ and $\beta' = -6$. We can construct the quadratic equation using the formula $x^2 - (\alpha' + \beta')x + \alpha' \beta' = 0$.

• Calculate the sum of the new roots: Sum = $(-2) + (-6) = -8$ Explanation: We add the two new roots.

• Calculate the product of the new roots: Product = $(-2) \times (-6) = 12$ Explanation: We multiply the two new roots.

• Substitute into the standard quadratic equation formula: $x^2 - (-8)x + 12 = 0$ $x^2 + 8x + 12 = 0$ Explanation: We replace the "sum of new roots" and "product of new roots" with their calculated values.

Common Mistakes & Tips

• Sign Errors: Be extremely careful with negative signs when applying Vieta's formulas and when calculating the sum and product of the roots.
• Fraction Arithmetic: Ensure accuracy when adding, subtracting, multiplying, and dividing fractions. A common mistake is not finding a common denominator.
• Understanding Vieta's Formulas: Make sure you correctly identify the coefficients $a$, $b$, and $c$ in the given quadratic equation before applying Vieta's formulas.

Summary

We used Vieta's formulas to find the coefficients $a$ and $b$ of the original quadratic equation. Then, we substituted these values into the expressions for the new roots and calculated their values. Finally, we used the sum and product of the new roots to form the new quadratic equation. The quadratic equation with roots -2 and -6 is $x^2 + 8x + 12 = 0$.

Final Answer

The final answer is $\boxed{x^2 + 8x + 12 = 0}$, which corresponds to option (A).