Quadratic Equations

We are given the roots of the quadratic equation as and . The sum of the roots is: The product of the roots is: The quadratic equation is given by: Substituting the sum and product of the roots: Multiplying through by 4 to clear the fraction: Thus, the correct answer is .

We use the identity: From the given quadratic equation, we know that: Substituting these into the identity: Thus, the correct answer is .

Let the two numbers be and . We are given the system of equations: We can solve these equations using the quadratic equation. Substitute into the second equation: Factoring the quadratic equation: Thus, and (or vice versa). Therefore, the correct answer is .

The sum of the roots of a quadratic equation is given by the formula: For the equation , we have , , and . Thus, the sum of the roots is: Thus, the correct answer is 3.

We use the fact that if 3 is a root of the equation, then it satisfies the equation: Simplifying: Thus, the correct answer is .

To find the range of for real and distinct roots, we proceed as follows:

1. For a quadratic equation , the roots are real and distinct if the discriminant .
2. Given: , so , , .
3. Calculate the discriminant:
4. Set the condition for real and distinct roots:
5. Solve the inequality:
6. Match with options: corresponds to option (A).

Thus, the correct answer is: