/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Solve. Some of your answers may ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 30

Solve. Some of your answers may involve \(i\). \(x^{2}+10 x+27=0\)

Short Answer

Expert verified
The roots are \(x = -5 \,\pm\, i\sqrt{2}\).

Step by step solution

01

Identify coefficients

The given quadratic equation is in the form of \(ax^{2}+bx+c=0\). Here, the coefficients are: \(a = 1\), \(b = 10\), \(c = 27\).
02

Calculate the discriminant

Use the discriminant formula, \(D = b^2 - 4ac\), to determine the type of roots.\[D = 10^2 - 4 \times 1 \times 27 = 100 - 108 = -8\]Since the discriminant is negative, the quadratic equation has complex roots.
03

Use the quadratic formula

The quadratic formula is \(x = \frac{-b \,\pm\, \sqrt{b^2-4ac}}{2a}\). Since \(D = -8\), we substitute into the formula:\[x = \frac{-10 \,\pm\, \sqrt{-8}}{2 \times 1} = \frac{-10 \,\pm\, \, 2i\frac{\sqrt{2}}{2}}{2}\]Therefore,\[x = \frac{-10 \,\pm\, 2i\sqrt{2}}{2}= -5 \,\pm\, i\sqrt{2}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

quadratic equations
A quadratic equation is a type of polynomial equation of degree two. It is expressed in the form of \[ax^2 + bx + c = 0\] where:
  • \(a\) is the coefficient of \(x^2\) (the quadratic term)
  • \(b\) is the coefficient of \(x\) (the linear term)
  • \(c\) is the constant term.
The key characteristic of a quadratic equation is that it produces a parabola when graphed on a coordinate plane. Quadratic equations can have three types of roots: real and distinct, real and repeated, or complex. To determine the type of roots, we use the discriminant.
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
  • \(b\) is the imaginary part, and it is multiplied by \(i\), the imaginary unit
The imaginary unit \(i\) is defined as \(\sqrt{-1}\), meaning \(i^2 = -1\). Complex numbers are essential for solving quadratic equations that do not have real roots. For instance, in the example quadratic equation, the discriminant was found to be negative, indicating complex roots. The roots are expressed as \(-5 \,\bullet\, i\sqrt{2}\). This means that there are two roots: \(-5 + i \sqrt{2}\) and \(-5 - i \sqrt{2}\).
discriminant
The discriminant is a part of the quadratic formula that determines the nature of the roots of the quadratic equation. It is given by the formula \[D = b^2 - 4ac\].Here's what the discriminant tells us:
  • If \(D > 0\), the quadratic equation has two distinct real roots.
  • If \(D = 0\), the quadratic equation has exactly one real root (a repeated root).
  • If \(D < 0\), the quadratic equation has two complex roots.
In the given problem, the coefficients of the quadratic equation are \(a = 1\), \(b = 10\), and \(c = 27\). The discriminant is calculated as:\[D = 10^2 - 4 \,\bullet\, 1 \,\bullet\, 27 = 100 - 108 = -8\].Since \(D\) is negative, the roots of this equation are complex. This is why we use \(i\) in our final answer.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Graph the following functions on \([-2 \pi, 2 \pi]\). a) \(y=2 \cos 2 t-1\) b) \(y=2 \cos (2[t+\pi / 6])-1\) c) How does the horizontal shift change the graph?

Use a calculator to evaluate the following trigonometric functions. $$ \sin (-2.04) $$

Use a calculator to find the radian measure of an acute angle whose trigonometric function is given. \(\tan t=0.11\)

Sketch the following angles. \(5 \pi / 4\)

Use a calculator to evaluate the following trigonometric functions. $$ \tan (-2.3 \pi) $$
See all solutions

Recommended explanations on Biology Textbooks

Communicable Diseases

Read Explanation

Cell Cycle

Read Explanation

Reproduction

Read Explanation

Plant Biology

Read Explanation

Biological Structures

Read Explanation

Heredity

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.