The two roots of a quadratic equation are `-7` and` `-3`, what will be the maximum value of this equation?

Options

A. 1
B. `-1`
C. 4
D. Cannot be determined uniquely

Answer

To find the maximum value of the quadratic equation with roots -7 and -3, we can follow these steps:

### Step 1: Write the quadratic equation using the roots
The general form of a quadratic equation with roots \( \alpha \) and \( \beta \) is given by:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Here, the roots are \( \alpha = -7 \) and \( \beta = -3 \).

### Step 2: Calculate the sum and product of the roots
First, we calculate the sum of the roots:
\[ \alpha + \beta = -7 + (-3) = -10 \]
Next, we calculate the product of the roots:
\[ \alpha \beta = (-7) \times (-3) = 21 \]

### Step 3: Substitute the values into the quadratic equation
Now, we substitute the sum and product of the roots into the general form:
\[ x^2 - (-10)x + 21 = 0 \]
This simplifies to:
\[ x^2 + 10x + 21 = 0 \]

### Step 4: Identify the coefficients
From the equation \( x^2 + 10x + 21 = 0 \), we identify the coefficients:
- \( a = 1 \)
- \( b = 10 \)
- \( c = 21 \)

### Step 5: Determine the nature of the quadratic
Since the coefficient of \( x^2 \) (which is \( a \)) is positive, the parabola opens upwards. Therefore, this quadratic function does not have a maximum value, but it has a minimum value.

### Step 6: Calculate the minimum value
The minimum value of a quadratic function can be found using the vertex formula:
\[ x = -\frac{b}{2a} \]
Substituting the values of \( b \) and \( a \):
\[ x = -\frac{10}{2 \times 1} = -5 \]

### Step 7: Substitute back to find the minimum value
Now we substitute \( x = -5 \) back into the quadratic equation to find the minimum value:
\[ f(-5) = (-5)^2 + 10(-5) + 21 \]
Calculating this gives:
\[ = 25 - 50 + 21 = -4 \]

### Conclusion
The minimum value of the quadratic equation is \(-4\), and since the parabola opens upwards, there is no maximum value.

### Final Answer
The maximum value of the equation cannot be determined.