Solve x² + 25 = 0
Solve the quadratic equation x² + 25 = 0 using the quadratic formula.
Solution:
Discriminant: Δ = -100 (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 = 25 (constant term)
Standard form: x² + 25 = 0
x² + 25 = 0
Step 2
Calculate the Discriminant
The discriminant determines the nature of the roots.
D = b² - 4ac
D = (0)² - 4(1)(25)
D = 0 - 100
D = -100
Since D = -100 < 0, the equation has two COMPLEX roots.
D = -100
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 = -100 < 0:
√D = √(100) × i = 10i
\sqrt{D} = 10i
Step 4
Calculate Complex Solutions
Real part: -b/(2a) = -0/2 = 0
Imaginary part: √|D|/(2a) = 10/2 = 5
The solutions are:
x₁ = 0 + 5i
x₂ = 0 - 5i
These are complex conjugates.
x = 0 \pm 5i
Step 5
Final Answer
x = 0 ± 5i