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Chapter 8: Problem 21

Solve. (Find all complex-number solutions.) \(p^{2}-p+1=0\)

Short Answer

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

Step by step solution

01

Identify the quadratic equation

The given problem is a quadratic equation in the form of \[ p^{2} - p + 1 = 0 \]
02

Use the Quadratic Formula

To find the solutions of the quadratic equation \( ax^{2} + bx + c = 0 \), we use the quadratic formula: \[ p = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]. Here, \( a = 1 \), \( b = -1 \), and \( c = 1 \).
03

Calculate the Discriminant

The discriminant of a quadratic equation is \( b^{2} - 4ac \). Substitute \( a = 1 \), \( b = -1 \), and \( c = 1 \) into the discriminant formula: \( (-1)^{2} - 4(1)(1) = 1 - 4 = -3 \). The discriminant is \( -3 \).
04

Find the Complex Solutions

Since the discriminant is negative, the equation has two complex solutions. Use the quadratic formula to find them: \[ p = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]. Substituting the values, we get: \[ p = \frac{-(-1) \pm \sqrt{-3}}{2(1)} = \frac{1 \pm \sqrt{-3}}{2} \]. Recall that \( \sqrt{-3} = i \sqrt{3} \). So the solutions are: \[ p = \frac{1 + i\sqrt{3}}{2} \] and \[ p = \frac{1 - i\sqrt{3}}{2} \].

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Key Concepts

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

quadratic equation
A quadratic equation is a polynomial equation of degree 2. In a general form, it looks like this:
\[ ax^2 + bx + c = 0 \]
where:
  • a, b, and c are constants
  • x is the variable
Quadratic equations can have up to two solutions. These solutions can be real or complex numbers. You can identify a quadratic equation by looking at its highest exponent, which is 2. For example:
\[ p^2 - p + 1 = 0 \]
is a quadratic equation because the highest power of the variable, p, is 2.
quadratic formula
The quadratic formula is a tool used to find the solutions of a quadratic equation. It works for any quadratic equation, whether the solutions are real or complex. The formula is:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
To use the quadratic formula, you plug in the values of a, b, and c from your quadratic equation. For example, in the equation \[ p^2 - p + 1 = 0 \], we have:
  • a = 1
  • b = -1
  • c = 1
To find the solutions, simply substitute these values into the quadratic formula.
discriminant
The discriminant is part of the quadratic formula that helps determine the nature of the solutions. It's given by:
\[ b^2 - 4ac \]
The value of the discriminant tells us if the solutions are real or complex:
  • If the discriminant is positive (>0), the quadratic equation has two distinct real solutions.
  • If the discriminant is zero (=0), the quadratic equation has one real solution (also called a repeated root).
  • If the discriminant is negative (<0), the quadratic equation has two complex solutions.
In our given equation \[ p^2 - p + 1 = 0 \], we calculate the discriminant as follows:
\[ (-1)^2 - 4(1)(1) = 1 - 4 = -3 \].
Since the discriminant is -3, the equation has two complex solutions.
complex numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form:
\[ a + bi \]
where:
  • a is the real part
  • bi is the imaginary part, with i representing the imaginary unit \( \sqrt{-1} \)
In the context of our quadratic equation \[ p^2 - p + 1 = 0 \], we found the solutions using the quadratic formula and obtained:
\[ p = \frac{1 \pm \sqrt{-3}}{2} \]
To express these as complex numbers, we use the fact that \( \sqrt{-1} = i \). Thus, \( \sqrt{-3} = i \sqrt{3} \), and the solutions become:
\[ p = \frac{1 + i\sqrt{3}}{2} \]
and
\[ p = \frac{1 - i\sqrt{3}}{2} \].
These are the complex-number solutions to our quadratic equation.

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