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Chapter 1: Problem 72

Solve each equation in the complex number system. $$ x^{2}-x+1=0 $$

Short Answer

Expert verified
The solutions are \( x = \frac{1 + \sqrt{3}i}{2} \) and \( x = \frac{1 - \sqrt{3}i}{2} \).

Step by step solution

01

- Identify the coefficients

Identify the coefficients of the quadratic equation given in the form: \( ax^2 + bx + c = 0 \)Here, the equation is \( x^2 - x + 1 = 0 \). Therefore, \( a = 1 \), \( b = -1 \), and \( c = 1 \).
02

- Apply the quadratic formula

The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute \( a = 1 \), \( b = -1 \), and \( c = 1 \) into the formula.
03

- Calculate the discriminant

Calculate the discriminant using the formula \( b^2 - 4ac \).\[ (-1)^2 - 4(1)(1) = 1 - 4 = -3 \]Since the discriminant is negative, the equation has complex solutions.
04

- Compute the complex solutions

Substitute the values and discriminant into the quadratic formula:\[ x = \frac{-(-1) \pm \sqrt{-3}}{2(1)} = \frac{1 \pm \sqrt{3}i}{2} \]Therefore, the solutions are:\[ x = \frac{1 + \sqrt{3}i}{2} \text{ and } x = \frac{1 - \sqrt{3}i}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic formula
The quadratic formula is essential for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula allows us to find the roots (solutions) of any quadratic equation whether the solutions are real or complex. The formula is:
\[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]
It's a straightforward plug-and-chug formula once you have identified the coefficients \( a \), \( b \), and \( c \) from your equation.
  • For our example equation \( x^2 - x + 1 = 0 \), we identified \( a = 1 \), \( b = -1 \), and \( c = 1 \).
  • We then substitute these values into the quadratic formula.
The quadratic formula will work for every quadratic equation, whether it has two real roots, one real root, or two complex roots.
complex solutions
Complex solutions occur when the quadratic equation's discriminant is negative. Complex numbers are expressed in the form \( a + bi \), where \( i \) is the imaginary unit and \( i^2 = -1 \).
When solving quadratic equations with the quadratic formula, if you encounter a negative discriminant (\( b^2 - 4ac < 0 \)), the square root of a negative number introduces the imaginary unit \( i \). For our equation:
  • We found that the discriminant \( b^2 - 4ac = -3 \) is negative.
  • This leads us to a complex solution.
  • By substituting the discriminant value back into the quadratic formula, we get: \( x = \frac{1 \, \pm \, \sqrt{-3}}{2} \).
  • Using the imaginary unit \( i \), we rewrite \( \, \sqrt{-3} = \, i\, \sqrt{3} \).
    • The final results are: \[ x = \frac{1 + \, \sqrt{3}i}{2} \text{ and } x = \frac{1 - \, \sqrt{3}i}{2} \]
discriminant
The discriminant is a crucial part of the quadratic formula and helps determine the nature of the roots of the quadratic equation. It is given by: \( b^2 - 4ac \). Based on its value:
  • If the discriminant is positive (> 0), the quadratic equation has two distinct real roots.
  • If the discriminant is zero (= 0), the equation has exactly one real root (a repeated root).
  • If the discriminant is negative (< 0), the equation has two complex conjugate roots.
For the problem \( x^2 - x + 1 = 0 \):
  • The discriminant is calculated as: \( (-1)^2 - 4(1)(1) = 1 - 4 = -3 \).
  • Since \( -3 \) is negative, it indicates that the roots are complex.
Understanding the discriminant helps you quickly determine the type of solutions you will get before even solving the equation.

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