What is the minimum value of the expression `x^(2)+8x+10`?

Answer

To find the minimum value of the expression \( x^2 + 8x + 10 \), we can follow these steps:

### Step 1: Identify the coefficients
The given expression is a quadratic equation of the form \( ax^2 + bx + c \).
Here, we have:
- \( a = 1 \)
- \( b = 8 \)
- \( c = 10 \)

### Step 2: Use the formula for the minimum value
The minimum value of a quadratic function \( ax^2 + bx + c \) can be found using the formula:
\[ \text{Minimum value} = c - \frac{b^2}{4a} \]
Substituting the values of \( a \), \( b \), and \( c \) into the formula:

### Step 3: Substitute the values into the formula
\[ \text{Minimum value} = 10 - \frac{8^2}{4 \cdot 1} \]

### Step 4: Calculate \( b^2 \) and \( 4a \)
First, calculate \( b^2 \):
\[ b^2 = 8^2 = 64 \]
Now calculate \( 4a \):
\[ 4a = 4 \cdot 1 = 4 \]

### Step 5: Substitute back into the formula
Now substitute these values back into the formula:
\[ \text{Minimum value} = 10 - \frac{64}{4} \]

### Step 6: Simplify the expression
Calculate \( \frac{64}{4} \):
\[ \frac{64}{4} = 16 \]
Now substitute this back:
\[ \text{Minimum value} = 10 - 16 \]

### Step 7: Final calculation
\[ \text{Minimum value} = -6 \]

Thus, the minimum value of the expression \( x^2 + 8x + 10 \) is **-6**.

---