Quadratic Equations

Step 1: Define the discriminant
The discriminant is: The value of determines the roots:
- : Two distinct real roots
- : Two real and equal roots (one repeated root)
- : Two complex conjugate roots

Step 2: Apply to the question
The question states the roots are real and equal, so:
Step 3: Check other options
- Option (1): : Gives complex roots, incorrect.
- Option (3): : Gives two distinct real roots, incorrect.
- Option (4): : This is the sum of the coefficients, which equals the value of the quadratic at . It’s unrelated to the condition for equal roots unless specific values are given. Incorrect.

Step 4: Conclusion
The condition for real and equal roots is .

To determine the correct relation among the coefficients of the quadratic equation when one root is double the other, follow these steps:
Let the roots be and . By Vieta's formulas, we know:

Also,
Substitute into :



Cross-multiply to clear fractions:

Dividing by , we get:

Thus, the correct relation is .

Applying the power rule to each term in :

Thus, the derivative is obtained by differentiating each term and combining them:

Therefore, the correct answer is .

A quadratic equation has real and equal roots when the discriminant is zero.

The discriminant formula is:

Given equation:

Apply the condition for equal roots: