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Chapter 3: Problem 3

Compute and plot the roots of the following quadratic equations: a. \(z^{2}+2 z+1=0\) b. \(z^{2}-2 z+1=0\) c. \(z^{2}=0\) For each equation, check that \(z_{1}+z_{2}=-\frac{b}{a}\) and \(z_{1} z_{2}=\frac{c}{a}\)

Short Answer

Expert verified
The roots are \(z = -1\), \(z = 1\), and \(z = 0\) respectively. The sum and product of roots for all equations are verified.

Step by step solution

01

Identify coefficients for equation (a)

For the equation \(z^{2} + 2z + 1 = 0\), the coefficients are: \(a = 1\), \(b = 2\), and \(c = 1\).
02

Find roots using the quadratic formula for equation (a)

Use the quadratic formula \[z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] to find the roots. Substitute \(a = 1\), \(b = 2\), and \(c = 1\): \[z = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-2 \pm \sqrt{0}}{2} = \frac{-2}{2} = -1\] Thus, the roots are \(z_1 = z_2 = -1\).
03

Verify sum and product of roots for equation (a)

Verify that \(z_1 + z_2 = -\frac{b}{a}\) and \(z_1 z_2 = \frac{c}{a}\). Since \(z_1 = z_2 = -1\), \[z_1 + z_2 = -1 + (-1) = -2 = -\frac{2}{1}\] and \[z_1 z_2 = (-1)(-1) = 1 = \frac{1}{1}\] Both conditions are satisfied.
04

Identify coefficients for equation (b)

For the equation \(z^{2} - 2z + 1 = 0\), the coefficients are: \(a = 1\), \(b = -2\), and \(c = 1\).
05

Find roots using the quadratic formula for equation (b)

Use the quadratic formula \[z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] to find the roots. Substitute \(a = 1\), \(b = -2\), and \(c = 1\): \[z = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{2 \pm \sqrt{0}}{2} = \frac{2}{2} = 1\] Thus, the roots are \(z_1 = z_2 = 1\).
06

Verify sum and product of roots for equation (b)

Verify that \(z_1 + z_2 = -\frac{b}{a}\) and \(z_1 z_2 = \frac{c}{a}\). Since \(z_1 = z_2 = 1\), \[z_1 + z_2 = 1 + 1 = 2 = -\frac{-2}{1}\] and \[z_1 z_2 = (1)(1) = 1 = \frac{1}{1}\] Both conditions are satisfied.
07

Identify coefficients for equation (c)

For the equation \(z^{2} = 0\), the coefficients are: \(a = 1\), \(b = 0\), and \(c = 0\).
08

Find roots for equation (c)

Since the equation is \(z^2 = 0\), we can solve directly: \(z = 0\). Thus, the roots are \(z_1 = z_2 = 0\).
09

Verify sum and product of roots for equation (c)

Verify that \(z_1 + z_2 = -\frac{b}{a}\) and \(z_1 z_2 = \frac{c}{a}\). Since \(z_1 = z_2 = 0\), \[z_1 + z_2 = 0 + 0 = 0 = -\frac{0}{1}\] and \[z_1 z_2 = (0)(0) = 0 = \frac{0}{1}\] Both conditions are satisfied.
10

Plot the roots

Plot the roots on a complex plane. For all equations, the roots are real numbers: \(z = -1\) for (a), \(z = 1\) for (b), and \(z = 0\) for (c).

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Key Concepts

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

roots of quadratic equations
When solving quadratic equations, we often need to find the roots, which are the values of the variable that satisfy the equation. This is key for understanding how the equation behaves and for solving real-world problems.

A quadratic equation can usually be written as: ax^2 + bx + c = 0.
The roots of this equation are the points where the equation intersects the x-axis. They can be real or complex numbers and are found using various methods, one of which is the quadratic formula.

The roots are often represented as \(z_1\) and \(z_2\). For instance:
  • Equation (a): \(z^2 + 2z + 1 = 0\) has roots \(z_1 = z_2 = -1\).
  • Equation (b): \(z^2 - 2z + 1 = 0\) has roots \(z_1 = z_2 = 1\).
  • Equation (c): \(z^2 = 0\) has roots \(z_1 = z_2 = 0\).
These roots help us understand the shape and position of the parabola defined by the quadratic equation on the graph.
quadratic formula
The quadratic formula is a powerful tool for finding the roots of a quadratic equation and is derived from completing the square. It's given by: \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
This formula works for any quadratic equation of the form \(ax^2 + bx + c = 0\) and helps us find both the roots, \(z_1\) and \(z_2\), all at once.
Here's how it's applied:
  • For equation (a): Apply \(a = 1\), \(b = 2\), and \(c = 1\), yielding \(z = \frac{-2 \pm \sqrt{4 - 4}}{2} = -1\).
  • For equation (b): Apply \(a = 1\), \(b = -2\), and \(c = 1\), resulting in \(z = \frac{2 \pm \sqrt{4 - 4}}{2} = 1\).
  • For equation (c): Apply \(a = 1\), \(b = 0\), and \(c = 0\), leading to \(z = \frac{0 \pm \sqrt{0}}{2} = 0\).
The term \(b^2 - 4ac\) under the square root is called the discriminant and determines the nature of the roots:
  • If it’s positive, we have two distinct real roots.
  • If it’s zero, we have exactly one real root, also called a repeated root.
  • If it’s negative, the roots are complex.
sum and product of roots
An important property of quadratic equations is found in their roots, specifically the sum and product. These can be derived from the coefficients of the quadratic equation without solving it completely.

Given \(ax^2 + bx + c = 0\), the sum of the roots \(z_1 + z_2\) is \(-\frac{b}{a}\), and the product \(z_1 \cdot z_2\) is \(\frac{c}{a}\).

Using these relations, we can quickly verify the roots:
  • For equation (a): \(z_1 = z_2 = -1\), so \(z_1 + z_2 = -2\), which matches \(-\frac{2}{1}\), and \(z_1 \cdot z_2 = 1\), matching \(\frac{1}{1}\).
  • For equation (b): \(z_1 = z_2 = 1\), thus \(z_1 + z_2 = 2\), matching \(-\frac{-2}{1}\), and \(z_1 \cdot z_2 = 1\), matching \(\frac{1}{1}\).
  • For equation (c): \(z_1 = z_2 = 0\), thus \(z_1 + z_2 = 0\), matching \(-\frac{0}{1}\), and \(z_1 \cdot z_2 = 0\), matching \(\frac{0}{1}\).
These verifications are a quick check to see if the solutions to the quadratic equation are correct and can save time during calculations.

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Most popular questions from this chapter

The following geometric codes represent complex numbers. Decode each by writing down the corresponding complex number \(z\) : a. \((0.7,-0.1)\) b. \((-1.0,0.5)\) c. \(1.6 \angle \pi / 8\) d. \(0.4 \angle 7 \pi / 8\)

Prove \((j)^{-1}=-j .\)

Call \(\mathrm{R}(\theta)\) the rotation matrix: $$ \mathrm{R}(\theta)=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] $$ Show that \(\mathrm{R}(-\theta)\) rotates by \((-\theta)\). What can you say about \(\mathrm{R}(-\theta) w\) when \(w=\mathrm{R}(\theta) z\) ?

Compute and plot the roots of the following quadratic equations: a. \(z^{2}+2 z+2=0\) b. \(z^{2}-2 z+2=0\) c. \(z^{2}+2=0\) For each equation, check that \(2 \operatorname{Re}\left[z_{1,2}\right]=-\frac{b}{a}\) and \(\left|z_{1,2}\right|^{2}=\frac{c}{a}\).

Begin with the complex number \(z_{1}=x+j y=r e^{j \theta}\). Compute the complex number \(z_{2}=j z_{1}\) in its Cartesian and polar forms. The complex number \(z_{2}\) is sometimes called perp \(\left(z_{1}\right) .\) Explain why by writing perp \(\left(z_{1}\right)\) as \(z_{1} e^{j \theta_{2}} .\) What is \(\theta_{2}\) ? Repeat this problem for \(z_{3}=-j z_{1}\).
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