Math Problem Statement
Solve the equation over the complex number system. X^2-2x+10=0
Solution
We are given the quadratic equation:
To solve this over the complex number system, we will use the quadratic formula:
For the given equation, we identify , , and .
Step 1: Calculate the discriminant
The discriminant is given by:
Since the discriminant is negative, the solutions will be complex.
Step 2: Apply the quadratic formula
Substitute , , and into the quadratic formula:
Step 3: Simplify the square root
The square root of is , where is the imaginary unit ():
Step 4: Simplify the expression
Now, divide both terms in the numerator by 2:
Final Answer:
The solutions are:
These are the two complex solutions.
Would you like further details or have any questions? Here are 5 related questions to consider:
- What is the significance of the discriminant in determining the nature of the roots?
- How would the solution differ if the discriminant were zero?
- Can you explain the role of the imaginary unit in complex numbers?
- How can you verify that is a solution to the original equation?
- What does it mean to solve an equation over the complex number system?
Tip: When the discriminant is negative, always expect the solutions to involve imaginary numbers.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Complex Numbers
Formulas
Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
Discriminant: Δ = b^2 - 4ac
Theorems
Quadratic formula
Nature of roots based on discriminant
Suitable Grade Level
Grades 9-12
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