Let f (x) = ax + bx + c = 0 (a ≠ 0) be such that f(-2)+f(3) = 0. If one of the roots of f (x) = 0 is a then the 3 times sum of the roots of f (x) = 0 is equal to:

We are given the quadratic equation \( f(x) = ax^2 + bx + c \), and we know that:

1. \( f(-2) + f(3) = 0 \)
2. One of the roots of \( f(x) = 0 \) is \( a \).

We need to determine the value of 3 times the sum of the roots.

### Step 1: Use \( f(-2) + f(3) = 0 \)

Let's calculate \( f(-2) \) and \( f(3) \).

\[
f(-2) = a(-2)^2 + b(-2) + c = 4a - 2b + c
\]
\[
f(3) = a(3)^2 + b(3) + c = 9a + 3b + c
\]

Now, add the two expressions:

\[
f(-2) + f(3) = (4a - 2b + c) + (9a + 3b + c)
\]
\[
f(-2) + f(3) = 4a + 9a - 2b + 3b + c + c = 13a + b + 2c
\]

Given that \( f(-2) + f(3) = 0 \), we have:

\[
13a + b + 2c = 0 \tag{1}
\]

### Step 2: Use the fact that one root is \( a \)

Let the roots of \( f(x) = 0 \) be \( \alpha \) and \( \beta \), where \( \alpha = a \).

The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[
\alpha + \beta = -\frac{b}{a}
\]

Thus, \( a + \beta = -\frac{b}{a} \), and we are asked to find 3 times the sum of the roots, which is:

\[
3(\alpha + \beta) = 3(-\frac{b}{a}) = -\frac{3b}{a}
\]

### Step 3: Conclusion

The value of 3 times the sum of the roots is:

\[
\boxed{-\frac{3b}{a}}
\]

Let me know if you'd like further details or clarifications.

### Follow-up questions:
1. What is the significance of the discriminant in a quadratic equation?
2. How can we solve quadratic equations using the quadratic formula?
3. What happens when the discriminant of a quadratic equation is zero?
4. How do we factorize a quadratic equation?
5. Can a quadratic equation have complex roots?

**Tip:** The sum and product of the roots of a quadratic equation are directly related to its coefficients, offering a shortcut to analyze them without solving explicitly!

Let f (x) = ax^2 + bx + c = 0 (a ≠ 0) be such that f(-2)+f(3) = 0. If one of the roots of f (x) = 0 is a, then the 3 times sum of the roots of f (x) = 0 is equal to:

Learn how to solve a quadratic equation where f(-2) + f(3) = 0 and one root is known. This step-by-step solution covers quadratic equations, the sum of roots, and key algebraic formulas, ideal for students in Grades 9-12.

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