Math Problem Statement

Solve the equation over the complex number system. X^2-2x+10=0

Solution

We are given the quadratic equation:

x22x+10=0x^2 - 2x + 10 = 0

To solve this over the complex number system, we will use the quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For the given equation, we identify a=1a = 1, b=2b = -2, and c=10c = 10.

Step 1: Calculate the discriminant

The discriminant is given by:

Δ=b24ac=(2)24(1)(10)=440=36\Delta = b^2 - 4ac = (-2)^2 - 4(1)(10) = 4 - 40 = -36

Since the discriminant is negative, the solutions will be complex.

Step 2: Apply the quadratic formula

Substitute a=1a = 1, b=2b = -2, and Δ=36\Delta = -36 into the quadratic formula:

x=(2)±362(1)=2±362x = \frac{-(-2) \pm \sqrt{-36}}{2(1)} = \frac{2 \pm \sqrt{-36}}{2}

Step 3: Simplify the square root

The square root of 36-36 is 6i6i, where ii is the imaginary unit (i=1i = \sqrt{-1}):

x=2±6i2x = \frac{2 \pm 6i}{2}

Step 4: Simplify the expression

Now, divide both terms in the numerator by 2:

x=1±3ix = 1 \pm 3i

Final Answer:

The solutions are:

x=1+3iorx=13ix = 1 + 3i \quad \text{or} \quad x = 1 - 3i

These are the two complex solutions.


Would you like further details or have any questions? Here are 5 related questions to consider:

  1. What is the significance of the discriminant in determining the nature of the roots?
  2. How would the solution differ if the discriminant were zero?
  3. Can you explain the role of the imaginary unit ii in complex numbers?
  4. How can you verify that x=1+3ix = 1 + 3i is a solution to the original equation?
  5. 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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