Question

Select for each of the following questions based on whether they have a real solution:

(A) x² = 49
(B) 2x² = -128
(C) 3x² - 16 = 32
(D) 2x² + 36 = 18
(E) x² + x + 1 = 0

Answer 1

To determine whether each equation has a real solution, we need to check if the discriminant is greater than or equal to 0. Equations (A), (B), and (C) have real solutions, while equations (D) and (E) do not.

Step-by-step explanation:

To determine whether each equation has a real solution, we need to check if the discriminant (b²-4ac) is greater than or equal to 0. If the discriminant is less than 0, it means the equation does not have a real solution.

(A) x² = 49:
Discriminant = 0² - 4(1)(-49) = 196, which is greater than 0. Therefore, there is a real solution.

(B) 2x² = -128:
Discriminant = 0² - 4(2)(-128) = 1024, which is greater than 0. Therefore, there is a real solution.

(C) 3x² - 16 = 32:
Discriminant = 0² - 4(3)(-48) = 576, which is greater than 0. Therefore, there is a real solution.

(D) 2x² + 36 = 18:
Discriminant = 0² - 4(2)(18-36) = -288, which is less than 0. Therefore, there is no real solution.

(E) x² + x + 1 = 0:
Discriminant = 1² - 4(1)(1) = -3, which is less than 0. Therefore, there is no real solution.