If f(x) =x²+3bx-4, then find the value of b so that f (5)=-9

To find the value of \( b \) such that \( f(5) = -9 \) for the function \( f(x) = x^2 + 3bx - 4 \), we can follow these steps:

1. **Substitute \( x = 5 \) into the function:**
   \[
   f(5) = 5^2 + 3b(5) - 4
   \]

2. **Simplify the equation:**
   \[
   f(5) = 25 + 15b - 4
   \]
   \[
   f(5) = 15b + 21
   \]

3. **Set the function equal to \(-9\) since \( f(5) = -9 \):**
   \[
   15b + 21 = -9
   \]

4. **Solve for \( b \):**
   \[
   15b = -9 - 21
   \]
   \[
   15b = -30
   \]
   \[
   b = \frac{-30}{15}
   \]
   \[
   b = -2
   \]

So, the value of \( b \) is \( \boxed{-2} \).

Would you like more details or have any questions about this problem? Here are some related questions to consider:

1. What would the value of \( b \) be if \( f(5) = 0 \)?
2. How would you find the value of \( b \) if the function was \( f(x) = x^2 - bx - 4 \) and \( f(5) = 11 \)?
3. What are the roots of the function \( f(x) = x^2 - 6x + 9 \)?
4. How can you determine the vertex of the parabola \( f(x) = x^2 + 3bx - 4 \)?
5. What is the value of \( b \) if \( f(x) = x^2 + 2bx - 4 \) and \( f(3) = -10 \)?

**Tip:** When solving quadratic equations, always simplify the expression step by step to avoid mistakes in algebraic manipulation.

Discover how to find the value of b for the quadratic function f(x) = x^2 + 3bx - 4 such that f(5) = -9. Follow a step-by-step solution methodically solving for b, involving algebraic manipulation and quadratic formula application. Suitable for high school students and above.

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