/*! 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 54 The given equation involves a po... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 54

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{2}-7=0\)

Short Answer

Expert verified
The real solutions are \(x = \sqrt{7}\) and \(x = -\sqrt{7}\).

Step by step solution

01

Identify the Equation

The given equation is a quadratic equation, and it is written as: \(x^{2}-7=0\). Our goal is to find all real solutions for \(x\).
02

Isolate the Quadratic Term

To solve for \(x\), first isolate the \(x^2\) term by adding 7 to both sides of the equation: \(x^{2} = 7\).
03

Solve for x

To find the values of \(x\), take the square root of both sides. This gives us two possible solutions: \(x = \pm\sqrt{7}\).
04

Confirm the Solutions

Verify the solutions by substituting them back into the original equation. For \(x = \sqrt{7}\): \((\sqrt{7})^2 = 7\). For \(x = -\sqrt{7}\): \((-\sqrt{7})^2 = 7\). Both satisfy the original equation.

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.

Solving Quadratics
Quadratic equations are expressions that follow the pattern: \( ax^2 + bx + c = 0 \). In these equations, \( a \), \( b \), and \( c \) are constants, with the variable \( x \) being raised to the second power or squared. Solving a quadratic equation involves finding the value of \( x \) that makes the equation true, known as the roots or solutions of the equation.
To solve, we often follow these steps:
  • Identify the quadratic equation and ensure all terms are on one side.
  • Isolate the quadratic term by performing algebraic operations.
  • Use methods like factoring, completing the square, or the quadratic formula to solve for \( x \).
The equation \( x^2 - 7 = 0 \) is already in a simplified form and can be solved easily by isolating the \( x^2 \) term. Once isolated, the next step often involves taking square roots to find the solution(s).
Square Roots
Taking square roots is a fundamental concept in solving quadratic equations, especially ones like \( x^2 = n \). When you "take the square root" of both sides of an equation, you're essentially finding a value that, when multiplied by itself, gives the desired number.
For example, if you have \( x^2 = 7 \), the square root of 7 will provide the solutions, \( x = \pm \sqrt{7} \).
  • The symbol \( \pm \) is important here. It indicates there are two possible solutions: one positive and one negative. This is because both \( (\sqrt{7})^2 \) and \( (-\sqrt{7})^2 \) equal 7.
  • Remember that the square root process introduces these two potential answers because a square is always non-negative, so both positive and negative roots are considered.
Understanding and applying the concept of square roots is crucial when solving quadratics directly through equating and square rooting as shown in the example.
Real Solutions
In the context of quadratic equations, real solutions refer to the values of \( x \) that are real numbers. Not all quadratic equations have real solutions; sometimes the roots are complex. However, when solving equations like \( x^2 = 7 \), the solutions are real because \( \sqrt{7} \) is a real number.
  • Real numbers include all the numbers on the number line: positive numbers, negative numbers, zero, and fractions or irrational numbers like \( \sqrt{7} \).
  • In our specific equation \( x^2 - 7 = 0 \), we obtain real solutions because we're dealing with positive quantities (squaring a number and setting it equal to a positive value).
Ensuring that we correctly identify whether solutions are real or complex is a key component of solving quadratic equations and understanding their scope. Verifying the solutions by substitution is a practical way to confirm they satisfy the original equation, as was done in the exercise, confirming real solutions \( x = \pm \sqrt{7} \).

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

Evaluate the expression and write the result in the form a bi. $$ (2+5 i)+(4-6 i) $$

Gas Mileage The gas mileage \(g\) (measured in mi/gal) for a particular vehicle, driven at \(v\) mi/h, is given by the formula \(g=10+0.9 v-0.01 v^{2},\) as long as \(v\) is between 10 \(\mathrm{mi} / \mathrm{h}\) and 75 \(\mathrm{mi} / \mathrm{h}\) . For what range of speeds is the vehicle's mileage 30 \(\mathrm{mi} / \mathrm{gal}\) or better?

Evaluate the expression and write the result in the form a bi. $$ 6 i-(4-i) $$

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \frac{1-\sqrt{-1}}{1+\sqrt{-1}} $$

What's Wrong Here? It is tempting to try to solve an in- equality as if it were an equation. For instance, we might try to solve \(1<3 / x\) by multiplying both sides by \(x\) , to get \(x<3,\) so the solution would be \((-\infty, 3) .\) But that's wrong; for example, \(x=-1\) lies in this interval but does not satisfy the original in- equality. Explain why this method doesn't work (think about the sign of \(x\) ). Then solve the inequality correctly.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.