`y=x^2-4,x+y=2`
Based on the system of equations above , what is the minimum value of the product xy ?

Options

A. 2
B. 0
C. `-5`
D. `-15`

Answer

To find the minimum value of the product \(xy\) based on the system of equations \(y = x^2 - 4\) and \(x + y = 2\), we can follow these steps:

### Step 1: Substitute \(y\) in the second equation
From the second equation, we can express \(y\) in terms of \(x\):
\[ y = 2 - x \]

### Step 2: Substitute \(y\) into the first equation
Now, we substitute \(y\) from Step 1 into the first equation:
\[ 2 - x = x^2 - 4 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 + x - 6 = 0 \]

### Step 4: Factor the quadratic equation
Next, we factor the quadratic equation:
\[ (x - 2)(x + 3) = 0 \]

### Step 5: Solve for \(x\)
Setting each factor to zero gives us the solutions for \(x\):
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

### Step 6: Find corresponding \(y\) values
Now we can find the corresponding \(y\) values for each \(x\):
1. For \(x = 2\):
\[ y = 2 - 2 = 0 \]
So, the pair is \((2, 0)\).

2. For \(x = -3\):
\[ y = 2 - (-3) = 5 \]
So, the pair is \((-3, 5)\).

### Step 7: Calculate the product \(xy\)
Now we calculate the product \(xy\) for both pairs:
1. For \((2, 0)\):
\[ xy = 2 \cdot 0 = 0 \]

2. For \((-3, 5)\):
\[ xy = -3 \cdot 5 = -15 \]

### Step 8: Determine the minimum value
The minimum value of the product \(xy\) from the two calculations is:
\[ \text{Minimum value of } xy = -15 \]

Thus, the minimum value of the product \(xy\) is \(-15\).

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