The quadratic equation whose roots are 3 and `-3` is .

Options

A. `x^(2)-9 = 0`
B. `x^(2)-3x - 3 =0`
C. `x^(2)-2x+2=0`
D. `x^(2)+9=0`

Answer

To find the quadratic equation whose roots are 3 and -3, we can follow these steps:

### Step 1: Identify the roots
The roots of the quadratic equation are given as \( \alpha = 3 \) and \( \beta = -3 \).

### Step 2: Use the sum and product of roots
The sum of the roots \( \alpha + \beta \) and the product of the roots \( \alpha \beta \) can be calculated as follows:
- Sum of the roots:
\[ \alpha + \beta = 3 + (-3) = 0 \]
- Product of the roots:
\[ \alpha \beta = 3 \times (-3) = -9 \]

### Step 3: Write the quadratic equation
The general form of a quadratic equation based on its roots is given by:
\[ x^2 - (\text{sum of roots}) \cdot x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - (0) \cdot x + (-9) = 0 \]
This simplifies to:
\[ x^2 - 9 = 0 \]

### Final Quadratic Equation
Thus, the quadratic equation whose roots are 3 and -3 is:
\[ x^2 - 9 = 0 \]

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