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

Question

Accepted Answer

To find the quadratic expression whose sum of roots and product of roots are -3 and 2 respectively, we can use the properties of quadratic equations. For a quadratic equation of the form $ ax^2 + bx + c = 0 $, the sum of the roots ($ \alpha + \beta $) is given by $ -b/a $, and the product of the roots ($ \alpha \beta $) is given by $ c/a $.

Given:
- Sum of roots ($ \alpha + \beta $) = -3
- Product of roots ($ \alpha \beta $) = 2

Since the quadratic expression is in the standard form $ x^2 + bx + c $, we have:
1. The sum of the roots is $ -b/a = -b/1 = -b $. Therefore, $ -b = -3 $, which implies $ b = 3 $.
2. The product of the roots is $ c/a = c/1 = c $. Therefore, $ c = 2 $.

Thus, the quadratic expression is $ x^2 + 3x + 2 $.

Now, let's analyze why Option 2 is correct and why the other options are incorrect:

1. **Option 1: $ x^2 - 3x - 2 $**
   - Sum of roots: $ -(-3) = 3 $ (not equal to -3)
   - Product of roots: $-2$ (not equal to 2)
   - This option does not satisfy either the sum or the product of the roots.

2. **Option 2: $ x^2 + 3x + 2 $**
   - Sum of roots: $-3$ (matches the given sum of roots)
   - Product of roots: $2$ (matches the given product of roots)
   - This option satisfies both the sum and the product of the roots, making it the correct answer.

3. **Option 3: $ x^2 - 5x + 2 $**
   - Sum of roots: $ -(-5) = 5 $ (not equal to -3)
   - Product of roots: $2$ (matches the product but not the sum)
   - This option does not satisfy the sum of the roots.

4. **Option 4: $ x^2 + 5x + 2 $**
   - Sum of roots: $-5$ (not equal to -3)
   - Product of roots: $2$ (matches the product but not the sum)
   - This option does not satisfy the sum of the roots.

In conclusion, Option 2 is the only choice that correctly satisfies both the sum and the product of the roots as given in the problem statement.