To determine the value of p for which the roots of the quadratic equation (p+3)x² + 2(p+3)x + 4 = 0 are equal, we need to follow the condition for equal roots.
Condition for Equal Roots
For a quadratic equation ax² + bx + c = 0, the roots are equal if the discriminant (D) is zero. The discriminant is given by:
D = b² - 4ac
In this case, we identify the coefficients:
- a = p + 3
- b = 2(p + 3)
- c = 4
Calculate the Discriminant
Now, we can express the discriminant:
- D = [2(p + 3)]² - 4(p + 3) * 4
Expand and Simplify
1. Calculate b²:
- [2(p + 3)]² = 4(p + 3)² = 4(p² + 6p + 9)
2. Calculate 4ac:
- 4(p + 3) * 4 = 16(p + 3)
Thus, the discriminant becomes:
- D = 4(p² + 6p + 9) - 16(p + 3)
Set the Discriminant to Zero
To find the value of p for equal roots:
- 4(p² + 6p + 9) - 16(p + 3) = 0
This simplifies to:
- 4p² + 24p + 36 - 16p - 48 = 0
Combine like terms:
- 4p² + 8p - 12 = 0
Factor and Solve
Now, divide the entire equation by 4:
- p² + 2p - 3 = 0
Factor the quadratic:
- (p + 3)(p - 1) = 0
This gives us the roots:
- p = -3 or p = 1
Conclusion
The values of p for which the roots of the quadratic equation are equal are:
- p = -3
- p = 1