To solve the quadratic equation x² + 5x + 1 = 0, we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
In this equation, a = 1, b = 5, and c = 1.
First, we calculate the discriminant (D):
D = b² – 4ac = 5² – 4(1)(1) = 25 – 4 = 21
Since D is positive, we have two distinct real solutions. Now, we can plug the values into the quadratic formula:
x = (-5 ± √21) / 2(1)
This gives us two solutions:
x₁ = (-5 + √21) / 2
x₂ = (-5 – √21) / 2
So, the solution set is:
{ (-5 + √21) / 2, (-5 – √21) / 2 }
In decimal form, these solutions approximately equal:
x₁ ≈ -0.79
x₂ ≈ -4.21
Thus, the complete solution set to the equation is the two values mentioned above.