Final answer:
The quadratic expressions x^2 + 4x + 21 and x^2 - 4x + 21 are similar in having the same leading coefficient and constant term. They differ in the sign of their linear term and neither has real roots due to negative discriminants.
Step-by-step explanation:
In analyzing the quadratic expressions x^2 + 4x + 21 and x^2 - 4x + 21, we can notice some similarities and differences regarding their structure and properties. To begin with, let's address the similarities and differences point by point:
- Both expressions have the same leading coefficient of 1 for the x^2 term.
- Both expressions have the same constant term of 21.
- As for the differences, they lie in the linear term, specifically the sign that precedes the '4x'. The first expression has a positive 4 as its coefficient, while the second has a negative 4.
Regarding the option c) that states both have real roots, we can ascertain this is incorrect by calculating the discriminant (b^2 - 4ac) for both equations:
- For x^2 + 4x + 21: the discriminant is (4^2) - 4(1)(21) = 16 - 84 = -68, which is negative.
- For x^2 - 4x + 21: the discriminant is (-4^2) - 4(1)(21) = 16 - 84 = -68, also negative.
This negative discriminant indicates that both expressions do not have real roots; instead, they have complex roots.