To find the solutions for the quadratic equation x² + 10x + 36 = 0, we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
In this equation, a = 1, b = 10, and c = 36. Now, we will calculate the discriminant (the part under the square root):
b² – 4ac = 10² – 4(1)(36)
= 100 – 144
= -44
Since the discriminant is negative (-44), this tells us that there are no real solutions to the equation. Instead, we have two complex (or imaginary) solutions.
Now, we can continue with the quadratic formula:
x = (-10 ± √(-44)) / 2(1)
This simplifies to:
x = (-10 ± 2i√11) / 2
Breaking this down further:
x = -5 ± i√11
Thus, the solutions for the equation x² + 10x + 36 = 0 are:
- x = -5 + i√11
- x = -5 – i√11
In conclusion, the equation has two complex solutions, and no real solutions exist.