the expression ` 2x ^2 + 4x +7` has minimum value m at `x= alpha ` . The ordered pair `( alpha , m) ` is

Options

A. (1, -5)
B. (-1, -5)
C. (-1,5)
D. (1, 5)

Answer

To find the ordered pair \((\alpha, m)\) where the expression \(2x^2 + 4x + 7\) has its minimum value, we can follow these steps:

### Step 1: Identify the quadratic expression
The given expression is:
\[ f(x) = 2x^2 + 4x + 7 \]

### Step 2: Rewrite the expression in vertex form
To find the minimum value, we can complete the square. Start by factoring out the coefficient of \(x^2\) from the first two terms:
\[ f(x) = 2(x^2 + 2x) + 7 \]

Next, complete the square for the expression inside the parentheses:
1. Take half of the coefficient of \(x\) (which is 2), square it: \((\frac{2}{2})^2 = 1\).
2. Add and subtract this square inside the parentheses:
\[ f(x) = 2(x^2 + 2x + 1 - 1) + 7 \]
\[ = 2((x + 1)^2 - 1) + 7 \]
\[ = 2(x + 1)^2 - 2 + 7 \]
\[ = 2(x + 1)^2 + 5 \]

### Step 3: Identify the vertex
The vertex form of the quadratic is:
\[ f(x) = 2(x + 1)^2 + 5 \]
From this form, we can see that the minimum value occurs when \((x + 1)^2\) is zero, which happens at:
\[ x + 1 = 0 \implies x = -1 \]
Thus, \(\alpha = -1\).

### Step 4: Find the minimum value \(m\)
Substituting \(\alpha = -1\) back into the expression to find \(m\):
\[ m = f(-1) = 2(-1 + 1)^2 + 5 = 2(0) + 5 = 5 \]

### Step 5: Write the ordered pair
The ordered pair \((\alpha, m)\) is:
\[ (-1, 5) \]

### Final Answer
The ordered pair \((\alpha, m)\) is:
\[ \boxed{(-1, 5)} \]