1. Key Concepts and Formulas

• Quadratic Expression: A quadratic expression can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$.
• Roots of a Quadratic Equation: If $\alpha$ and $\beta$ are the roots of the quadratic equation $f(x) = 0$, then $f(x)$ can be written as $f(x) = k(x - \alpha)(x - \beta)$, where $k$ is a non-zero constant.
• Relationship between roots and coefficients: For a quadratic equation $ax^2+bx+c=0$, the sum of roots is $-\frac{b}{a}$ and the product of roots is $\frac{c}{a}$.

2. Step-by-Step Solution

Step 1: Formulate the quadratic expression using the given root.

We are given that $f(x)$ is a quadratic expression and one of its roots is $-1$. Let the other root be $\alpha$. Using the general form of a quadratic expression with roots, we can write: $f(x) = k(x - \alpha)(x - (-1))$ $f(x) = k(x - \alpha)(x + 1)$

Explanation: We use $\alpha$ to represent the unknown other root. The constant $k$ scales the quadratic expression. Since we are looking for the other root of $f(x)=0$, and we have a condition $f(1)+f(2)=0$ which implies $k$ will divide out, we can simplify our calculations by assuming $k=1$. This assumption does not affect the values of the roots. Expanding the expression with $k=1$: $f(x) = (x - \alpha)(x + 1)$ $f(x) = x^2 + x - \alpha x - \alpha$ $f(x) = x^2 + (1 - \alpha)x - \alpha$

Step 2: Utilize the given condition $f(1) + f(2) = 0$.

We need to find the values of $f(1)$ and $f(2)$ by substituting $x=1$ and $x=2$ into our derived expression for $f(x)$.

First, calculate $f(1)$: Substitute $x=1$ into $f(x) = (x - \alpha)(x + 1)$: $f(1) = (1 - \alpha)(1 + 1)$ $f(1) = (1 - \alpha)(2)$ $f(1) = 2 - 2\alpha$

Explanation: We evaluate the quadratic at $x=1$ to use the first part of the given condition.

Next, calculate $f(2)$: Substitute $x=2$ into $f(x) = (x - \alpha)(x + 1)$: $f(2) = (2 - \alpha)(2 + 1)$ $f(2) = (2 - \alpha)(3)$ $f(2) = 6 - 3\alpha$

Explanation: Similarly, we evaluate the quadratic at $x=2$ for the second part of the condition.

Now, apply the condition $f(1) + f(2) = 0$: $(2 - 2\alpha) + (6 - 3\alpha) = 0$

Explanation: We substitute the expressions for $f(1)$ and $f(2)$ into the given sum condition, setting it equal to zero to form an equation in terms of $\alpha$.

Step 3: Solve for the unknown root $\alpha$.

Combine like terms in the equation from Step 2: $2 - 2\alpha + 6 - 3\alpha = 0$ $8 - 5\alpha = 0$ Now, isolate $\alpha$: $5\alpha = 8$ $\alpha = \frac{8}{5}$

Explanation: This is a simple linear equation. We rearrange the terms to solve for $\alpha$, which represents the other root of the quadratic equation.

3. Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when expanding and substituting. A small sign error can lead to an incorrect answer.
• Assuming k=1: When given conditions such as $f(1) + f(2) = 0$, the leading coefficient 'k' can usually be assumed to be 1, as it will cancel out when solving for the roots. However, remember that k cannot be zero.
• Understanding Roots: Remember the fundamental relationship between the roots of a quadratic equation and its factored form.

4. Summary

By utilizing the relationship between the roots and the general form of a quadratic expression, and systematically applying the given condition $f(1) + f(2) = 0$, we determined the unknown root. We first expressed the quadratic in terms of its known root (-1) and the unknown root (α). Then, by using the condition $f(1)+f(2)=0$, we formed an equation in terms of α and solved for it. The other root of the quadratic expression $f(x) = 0$ is $\frac{8}{5}$.

5. Final Answer

The final answer is $\boxed{\frac{8}{5}}$, which corresponds to option (D).