How many solutions does 3x squared plus 4x plus 5 = 0 have, and of what type?

1. The number and type of solutions follow from the discriminant b squared minus 4ac. 2. Read off a as 3, b as 4 and c as 5. 3. Squaring b gives 16. 4. Working out 4ac gives 4 times 3 times 5, which is 60. 5. So the discriminant is 16 minus 60, which is negative 44. 6. A negative discriminant means the equation has 2 complex solutions and no real ones. 7. Answering two real solutions would need a positive discriminant. 8. Answering one real solution would need a discriminant of exactly 0. 9. Complex solutions of a real quadratic always arrive in a pair, so one complex solution is impossible. _Source: OpenStax Intermediate Algebra (CC BY 4.0), section 9.3 Solve Quadratic Equations Using the Quadratic Formula_