52.	The minimum value of the function y = x2 + bx + 16 is –9, where b is a constant. Find the two possible values of b.

To find the two possible values of \( b \) for which the minimum value of the function \( y = x^2 + bx + 16 \) is \(-9\), we need to follow these steps:

1. **Identify the vertex form of the quadratic function:**
   The given quadratic function is \( y = x^2 + bx + 16 \). 

2. **Find the vertex:**
   The vertex form of a quadratic function \( y = ax^2 + bx + c \) is given by:
   \[ x = -\frac{b}{2a} \]
   Here, \( a = 1 \) and \( c = 16 \), so:
   \[ x = -\frac{b}{2} \]

3. **Substitute the vertex x-coordinate back into the function:**
   Substitute \( x = -\frac{b}{2} \) back into \( y = x^2 + bx + 16 \):
   \[ y = \left(-\frac{b}{2}\right)^2 + b\left(-\frac{b}{2}\right) + 16 \]

4. **Simplify the expression:**
   \[ y = \frac{b^2}{4} - \frac{b^2}{2} + 16 \]
   \[ y = \frac{b^2}{4} - \frac{2b^2}{4} + 16 \]
   \[ y = -\frac{b^2}{4} + 16 \]

5. **Set the minimum value of the function to -9 and solve for \( b \):**
   Given that the minimum value is \(-9\):
   \[ -\frac{b^2}{4} + 16 = -9 \]

6. **Solve for \( b \):**
   \[ -\frac{b^2}{4} + 16 = -9 \]
   \[ -\frac{b^2}{4} = -9 - 16 \]
   \[ -\frac{b^2}{4} = -25 \]
   \[ \frac{b^2}{4} = 25 \]
   \[ b^2 = 100 \]
   \[ b = \pm 10 \]

So, the two possible values of \( b \) are:
\[ b = 10 \]
\[ b = -10 \]

If you have any questions or need further details, feel free to ask. Here are some questions you might want to explore next:

1. How do you find the maximum value of a quadratic function?
2. How does the sign of \(a\) in a quadratic function \(y = ax^2 + bx + c\) affect the direction of the parabola?
3. What are the roots of the quadratic equation \( y = x^2 + bx + 16 \) for \( b = 10 \)?
4. How do you complete the square for the function \( y = x^2 + bx + 16 \)?
5. What is the axis of symmetry for the quadratic function \( y = x^2 + bx + 16 \)?

**Tip:** Remember that the vertex form of a quadratic function can be very helpful in identifying the maximum or minimum value of the function, especially when the function is not easily factorable.

Discover how to find the minimum value of the quadratic function y = x^2 + bx + 16 by identifying the vertex form and solving for the constant b. Detailed steps and solutions provided. Suitable for high school students.

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