/*! 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 106 Factor completely. $$7 x^{4}+3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 106

Factor completely. $$7 x^{4}+34 x^{2}-5$$

Short Answer

Expert verified
The factorised form of the equation \(7x^4 + 34x^2 - 5\) is \( (x - i)(x + i)(x - sqrt(5)/sqrt(7))(x + sqrt(5)/sqrt(7))\).

Step by step solution

01

Identify the Concealed Quadratic Form

Start by rewriting the equation but replacing \(x^2\) with a single variable, say \(y\), so our equation now reads \(7y^2 + 34y - 5\).
02

Factor Using Quadratic Formula

Now apply the quadratic formula \((-b 卤 sqrt(b^2 - 4ac))/2a\) to find values of \(y\). Here \(a = 7\), \(b = 34\), \(c = -5\). Which leads to, \(y = (-34 卤 sqrt((34)^2 - 4*7*(-5)))/2*7\) or \(y = (-34 卤 sqrt(1156 + 140))/14 \) or \(y = (-34 卤 sqrt(1296))/14 \) thus \(y = -1\) or \(y = 5/7\).
03

Re-substitute \(x^2\)

Once the values of \(y\) are found, substitute back \(y\) with \(x^2\) as before, that gives \(x^2 = -1\) which implies \(x = i, -i\). Also, \(x^2 = 5/7\) that implies \(x = (sqrt(5))/sqrt(7) , -sqrt(5)/sqrt(7)\).
04

Write Final Factorised Form

The final factorised form of the original equation is \( (x - i)(x + i)(x - sqrt(5)/sqrt(7))(x + sqrt(5)/sqrt(7))\) after the process of substitution.

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 Formula
The quadratic formula is a powerful tool used to determine the solutions of quadratic equations of the form \( ax^2 + bx + c = 0 \). Although this looks complex, the quadratic formula makes solving equations much simpler. This formula is given by:
  • \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)
To use the formula, you need the coefficients \( a \), \( b \), and \( c \) from the quadratic equation. In the context of this exercise:
  • \( a = 7 \), \( b = 34 \), and \( c = -5 \)
By plugging these values into the formula, you can find the values of \( y \) 鈥� solutions to the quadratic equation 鈥� which turn out to be \( y = -1 \) and \( y = \frac{5}{7} \).
After this, these solutions are crucial for re-substitution and factoring polynomial expressions efficiently.
Complex Numbers
Complex numbers become essential when solving equations that don鈥檛 have real number solutions. When taking the square root of a negative number, like in the step where you set \( x^2 = -1 \), you encounter complex numbers.
In mathematics,
  • \( i \) is used to denote \( \sqrt{-1} \)
Thus, solving \( x^2 = -1 \) gives the roots \( x = i \) and \( x = -i \). These solutions are complex because they involve the imaginary unit \( i \).
In real-world terms, although these solutions may represent none of the physically measurable quantities, they are crucial in advanced mathematics and engineering.
Hence, awareness of complex numbers expands your ability to solve a broader array of equations.
Algebraic Substitution
Algebraic substitution involves replacing one component of an equation with another equivalent expression, simplifying the problem. This technique allows one to temporarily reduce complexity.
In this problem, the substitution was made by setting \( y = x^2 \). This transforms the quartic equation \( 7x^4 + 34x^2 - 5 \) into a quadratic form: \( 7y^2 + 34y - 5 \).
  • With \( y = x^2 \), solving becomes more straightforward, as it transitions to a familiar quadratic equation.
  • After solving for \( y \), you simply substitute back \( x^2 \) in place of \( y \).
This approach demonstrates how complex polynomial expressions can be tackled by converting them into simpler forms. It aids in easing the factorization process which might look daunting initially.

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

Find all integers \(b\) so that the trinomial can be factored. $$x^{2}+4 x+b$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I factored \(4 x^{2}-100\) completely and obtained \((2 x+10)(2 x-10)\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The model \(P=-0.18 n+2.1\) describes the number of pay phones, \(P,\) in millions, \(n\) years after \(2000,\) so I have to solve a linear equation to determine the number of pay phones in 2010

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x,\) is modeled by \(|x-20| \leq 5\)

List all numbers that must be excluded from the domain of each rational expression. $$\frac{7}{2 x^{2}-8 x+5}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.