Solve x² + 2x + 5 = 0
Solve the quadratic equation x² + 2x + 5 = 0 using the quadratic formula.
Solution:
Discriminant: Δ = -16 (2 complex roots)
Step-by-Step Solution
Step 1
Identify the Standard Form
A quadratic equation has the form: ax² + bx + c = 0
From the given equation:
• a = 1 (coefficient of x²)
• b = 2 (coefficient of x)
• c = 5 (constant term)
Standard form: x² + 2x + 5 = 0
x² + 2x + 5 = 0
Step 2
Calculate the Discriminant
The discriminant determines the nature of the roots.
D = b² - 4ac
D = (2)² - 4(1)(5)
D = 4 - 20
D = -16
Since D = -16 < 0, the equation has two COMPLEX roots.
D = -16
Step 3
Apply the Quadratic Formula for Complex Roots
Since D < 0, we have complex roots.
The quadratic formula gives:
x = (-b ± √D) / 2a
With D = -16 < 0:
√D = √(16) × i = 4i
\sqrt{D} = 4i
Step 4
Calculate Complex Solutions
Real part: -b/(2a) = -2/2 = -1
Imaginary part: √|D|/(2a) = 4/2 = 2
The solutions are:
x₁ = -1 + 2i
x₂ = -1 - 2i
These are complex conjugates.
x = -1 \pm 2i
Step 5
Final Answer
x = -1 ± 2i