Find the quadratic expression, whose sum of roots and product of roots are -3 and 2 respectively:

Options

A. A)`x^2-3x-2`
B. B)`x^2+3x+2`
C. C)`x^2-5x+2`
D. D)`x^2+5x+2`

Answer

To find the quadratic expression whose sum of roots is -3 and product of roots is 2, we can follow these steps:

### Step 1: Understand the Form of a Quadratic Equation
A quadratic equation can be represented in the standard form:
\[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants.

### Step 2: Use the Relationships of Roots
For a quadratic equation, the sum of the roots (let's denote them as \( \alpha \) and \( \beta \)) can be expressed as:
\[ \alpha + \beta = -\frac{b}{a} \]
And the product of the roots is given by:
\[ \alpha \cdot \beta = \frac{c}{a} \]

### Step 3: Substitute the Given Values
From the problem, we know:
- Sum of roots \( \alpha + \beta = -3 \)
- Product of roots \( \alpha \cdot \beta = 2 \)

Using the sum of roots:
\[ -\frac{b}{a} = -3 \]
This implies:
\[ \frac{b}{a} = 3 \]

Using the product of roots:
\[ \frac{c}{a} = 2 \]

### Step 4: Assume \( a = 1 \) for Simplicity
To simplify our calculations, we can assume \( a = 1 \). Therefore:
- From \( \frac{b}{a} = 3 \), we get \( b = 3 \).
- From \( \frac{c}{a} = 2 \), we get \( c = 2 \).

### Step 5: Write the Quadratic Expression
Substituting \( a \), \( b \), and \( c \) back into the quadratic equation gives us:
\[ x^2 + 3x + 2 = 0 \]

### Step 6: Final Expression
Thus, the required quadratic expression is:
\[ x^2 + 3x + 2 \]