/*! 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 47 Find the real solutions of each ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 47

Find the real solutions of each equation. $$ x^{6}+7 x^{3}-8=0 $$

Short Answer

Expert verified
The real solutions are x = 1 and x = -2.

Step by step solution

01

- Identify a substitution

Notice that the equation can be simplified using a substitution. Let鈥檚 set: y = x^3This transforms the original equation into a quadratic form.
02

- Substitute and simplify

Substitute y = x^3 into the given equation: y^2 + 7y - 8 = 0Now, solve the quadratic equation.
03

- Solve the quadratic equation

Factorize the quadratic equation: y^2 + 7y - 8 = 0 y^2 + 8y - y - 8 = 0 y(y + 8) - 1(y + 8) = 0 y = 1 or y = -8
04

- Substitute back to original variable

Substitute back y = x^3: x^3 = 1 or x^3 = -8
05

- Find the real solutions

Solve for x: x^3 = 1 gives x = 1 x^3 = -8 gives x = -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.

Polynomial Substitution
Polynomial substitution is a useful technique for simplifying complex polynomial equations. It involves replacing a part of the polynomial with a new variable, making it easier to solve. For example, in the equation \(x^{6} + 7x^{3} - 8 = 0\), we notice that the term \(x^{6}\) can be expressed as \((x^{3})^{2}\). By setting \(y = x^{3}\), the original equation simplifies to \(y^{2} + 7y - 8 = 0\). This transformation makes the polynomial easier to handle, as we convert it into a quadratic form, which is simpler to solve.
Quadratic Equation
A quadratic equation is a second-degree polynomial equation of the form \(ax^{2} + bx + c = 0\). In our simplified equation, \(y^{2} + 7y - 8 = 0\), the coefficients are:
  • a = 1
  • b = 7
  • c = -8
. Solving quadratic equations can be done using various methods such as factoring, completing the square, or the quadratic formula. For our example, we used factoring.
Real Solutions
Real solutions are the values of the variable that satisfy the equation and are real numbers (not imaginary or complex numbers). After simplifying our polynomial to a quadratic equation, we found the solutions to be \(y = 1\) and \(y = -8\). Since \(y\) represents \(x^{3}\), we then need to solve \(x^{3} = 1\) and \(x^{3} = -8\) to find the real solutions. For \(x^{3} = 1\), the real solution is \(x = 1\). For \(x^{3} = -8\), the real solution is \(x = -2\). Thus, the real solutions to the original equation \(x^{6} + 7x^{3} - 8 = 0\) are \(x = 1\) and \(x = -2\).
Factoring
Factoring is the process of breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. For the quadratic equation \(y^{2} + 7y - 8 = 0\), we factorize it by expressing it as a product of two binomials: \(y^{2} + 7y - 8 = (y + 8)(y - 1) = 0\). Setting each factor equal to zero gives us the solutions:
  • \(y + 8 = 0 \) thus \( y = -8 \)
  • \(y - 1 = 0 \) thus \( y = 1 \)
. Factoring is a key technique in solving polynomial equations, especially when they can be transformed into quadratic equations, as shown in this problem.

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

Dimensions of a Window The area of the opening of a rectangular window is to be 143 square feet. If the length is to be 2 feet more than the width, what are the dimensions?

Find the real solutions, if any, of each equation. Use any method. $$ \frac{3 x}{x+2}+\frac{1}{x-1}=\frac{4-7 x}{x^{2}+x-2} $$

Find all the complex solutions of \(z^{6}+28 z^{3}+27=0\).

Find the real solutions, if any, of each equation. Use any method. $$ 2=y+6 y^{2} $$

Exercising Sue wants to lose weight. For healthy weight loss, the American College of Sports Medicine (ACSM) recommends 200 to 300 minutes of exercise per week. For the first six days of the week, Sue exercised 40,45 , \(0,50,25,\) and 35 minutes. How long should Sue exercise on the seventh day in order to stay within the ACSM guidelines?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

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.