Which of the following equations has the sum of its roots as 3?

To determine which equation has the sum of its roots equal to 3, we need to recall the fundamental relationship between the coefficients of a quadratic equation and its roots. For a standard quadratic equation of the form:

a x^2 + b x + c = 0

The sum of the roots, denoted as r₁ and r₂, can be calculated using the formula:

ext{Sum of roots} = r₁ + r₂ = -rac{b}{a}

Here, a represents the coefficient of x², and b is the coefficient of x. Therefore, to find an equation where the sum of the roots is equal to 3, we can set up the equation:

- rac{b}{a} = 3

This leads us to:

b = -3a

Now, we can create various equations by choosing different values for a. For example:

• If we let a = 1, then b = -3. Thus, our equation becomes:

ext{Equation: } x^2 - 3x + c = 0

• If c is any real number, the sum of the roots will still equal 3.
• Alternatively, with a = 2, then b = -6, leading to:

ext{Equation: } 2x^2 - 6x + c = 0

• Again, the sum of the roots remains 3 for any real number c.

As you can see, there are multiple equations that satisfy the condition of having the sum of their roots as 3 by appropriately selecting a and b.

Quick Tip: Remember that to find the sum of the roots of a quadratic equation, simply use the relation -b/a. This can help you quickly check if your chosen equation meets the criteria!