1.What is the nature of th roots of f(x) = x^2 + 4x+16-I think it is two conjugate complex roots because it is -2+-2isquare root 3?
2.Nature of roots of f(x) = X^2 + 2x+1 it would be one double root,correct?
I get confused with doing these sometimes, just wanted to verify this is correct-thanks for checking
correct for both
To determine the nature of the roots for these quadratic equations, we can analyze their discriminants. The discriminant is a mathematical term that helps us determine whether the roots are real, complex, or repeated.
1. For the equation f(x) = x^2 + 4x + 16, the discriminant is given by b^2 - 4ac, where a = 1, b = 4, and c = 16. Plugging the values into the formula, we have:
Discriminant = (4^2) - 4(1)(16)
= 16 - 64
= -48
Since the discriminant is negative (-48 < 0), the roots of the equation f(x) = x^2 + 4x + 16 are complex conjugates. You correctly identified that the roots will be in the form of -2 ± 2i√3.
2. For the equation f(x) = x^2 + 2x + 1, we consider the discriminant again. Here, a = 1, b = 2, and c = 1. Plugging in the values, we have:
Discriminant = (2^2) - 4(1)(1)
= 4 - 4
= 0
Since the discriminant is zero (0), the roots of the equation f(x) = x^2 + 2x + 1 are repeated or have one double root. Therefore, you are correct in identifying that this equation has one double root.
Understanding the discriminant and its relationship to the nature of the roots can aid in determining the behavior of quadratic equations and confirming your answers.