/*! 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 60 Solve. $$ 8 n^{3}+10 n^{2}+3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 60

Solve. $$ 8 n^{3}+10 n^{2}+3 n=0 $$

Short Answer

Expert verified
The solutions are \( n = 0 \), \( n = -\frac{1}{2} \), and \( n = -\frac{3}{4} \).

Step by step solution

01

Factor Out Common Terms

First, notice that each term in the equation has a common factor of \(n\). So, factor \(n\) out of the equation: \[ n(8n^2 + 10n + 3) = 0 \]
02

Solve the Factored Equation

Set each factor equal to zero and solve for \(n\). First factor: \[ n = 0 \]Second factor: \[ 8n^2 + 10n + 3 = 0 \]
03

Solve the Quadratic Equation

Now, solve the quadratic equation \( 8n^2 + 10n + 3 = 0 \) using the quadratic formula: \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this case, \(a = 8\), \(b = 10\), and \(c = 3\). Substitute these values in:\[ n = \frac{-10 \pm \sqrt{10^2 - 4(8)(3)}}{2(8)} \]Simplify inside the square root:\[ n = \frac{-10 \pm \sqrt{100 - 96}}{16} \]This simplifies to:\[ n = \frac{-10 \pm 2}{16} \]
04

Simplify the Solutions

Now simplify the expression by splitting it into two separate solutions:\[ n = \frac{-10 + 2}{16} = \frac{-8}{16} = -\frac{1}{2} \]\[ n = \frac{-10 - 2}{16} = \frac{-12}{16} = -\frac{3}{4} \]
05

Combine All Solutions

List all solutions from the previous steps: \( n = 0 \), \( n = -\frac{1}{2} \), and \( n = -\frac{3}{4} \)

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.

Factoring Polynomials
Factoring polynomials is a fundamental technique for solving higher-degree equations. It involves expressing the polynomial as a product of simpler polynomials. To solve the cubic equation in the exercise, we started by noticing that every term had a common factor of \(n\). Factoring that out, we transformed:
  • \(8n^3 + 10n^2 + 3n = 0\) into \(n(8n^2 + 10n + 3) = 0\)
This factorization simplifies the task by breaking down the original cubic equation into a simpler quadratic equation plus a linear term. The term \(n = 0\) is straightforward, while the quadratic equation requires further steps.
Quadratic Formula
The quadratic formula is an essential tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\).
  • Given the quadratic equation \(8n^2 + 10n + 3 = 0\), we apply the quadratic formula: \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
  • Here, \(a = 8\), \(b = 10\), and \(c = 3\).
Substituting these values into the formula, we get: \[ n = \frac{-10 \pm \sqrt{100 - 96}}{16} \]Simplifying inside the square root:\[ n = \frac{-10 \pm \sqrt{4}}{16} \]We further simplify to:\[ n = \frac{-10 \pm 2}{16} \]This results in the two solutions for the quadratic equation.
Algebraic Solutions
Combining algebraic techniques helps in solving polynomial equations. Starting with the cubic equation:\(8n^3 + 10n^2 + 3n = 0\), we factor out the common term \(n\), leading to:\(n(8n^2 + 10n + 3) = 0\).
  • Setting each factor to zero, we solve the simpler equations: \(n = 0\), and \(8n^2 + 10n + 3 = 0\).
For \(8n^2 + 10n + 3 = 0\), applying the quadratic formula yields:
  • \(n = \frac{-10 + 2}{16} = -\frac{1}{2}\)
  • \(n = \frac{-10 - 2}{16} = -\frac{3}{4}\)
Combining these, the final solutions are:
  • \(n = 0\)
  • \(n = -\frac{1}{2}\)
  • \(n = -\frac{3}{4}\)
By leveraging factoring and the quadratic formula, we resolved the cubic equation algebraically.

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

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ -64+m^{2} $$

Write an equivalent expression by factoring out the smallest power of \(x\) in each of the following. $$ x^{-6}+x^{-9}+x^{-3} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ -80 a^{6}+45 $$

Ashlee factors \(8 x^{2} y-10 x y^{2}\) as $$ 2 x y \cdot 4 x-2 x y \cdot 5 y $$ Is this the factorization of the polynomial? Why or why not?

Determine whether each of the following is a difference of squares. $$ -12+25 t^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.