Solve x² + 2 = 0
Solve the quadratic equation x² + 2 = 0 using the quadratic formula.
Solution:
Discriminant: Δ = -8 (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 = 0 (coefficient of x)
• c = 2 (constant term)
Standard form: x² + 2 = 0
x² + 2 = 0
Step 2
Calculate the Discriminant
The discriminant determines the nature of the roots.
D = b² - 4ac
D = (0)² - 4(1)(2)
D = 0 - 8
D = -8
Since D = -8 < 0, the equation has two COMPLEX roots.
D = -8
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 = -8 < 0:
√D = √(8) × i = 2.828427i
\sqrt{D} = 2.828427i
Step 4
Calculate Complex Solutions
Real part: -b/(2a) = -0/2 = 0
Imaginary part: √|D|/(2a) = 2.828427/2 = 1.414214
The solutions are:
x₁ = 0 + 1.414214i
x₂ = 0 - 1.414214i
These are complex conjugates.
x = 0 \pm 1.414214i
Step 5
Final Answer
x = 0 ± 1.414214i