/*! 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 72 Solve the equation by extracting... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 72

Solve the equation by extracting square roots. $$ x^{2}=32 $$

Short Answer

Expert verified
The solution to the equation is x = \(\pm 4\sqrt{2}\).

Step by step solution

01

Squaring both sides

Firstly, perform a square root operation on both sides of the equation. So, the equation transforms into: \( x = \sqrt{32} \)
02

Simplifying the square root

32 can be broken down into 16 * 2, and the square root of 16 is 4. Thus, \( x = \sqrt{16 * 2} \) simplifies to \( x = 4\sqrt{2} \)
03

Considering both positive and negative roots

When performing a square root operation, it's important to consider both the positive and negative roots, as both are valid solutions for the square equation. So, x can be both \(4\sqrt{2}\) and \(-4\sqrt{2}\).

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.

Solving Quadratic Equations
Understanding how to solve quadratic equations is a foundational skill in algebra. A quadratic equation is a second-order polynomial equation in a single variable x with a <>non-zero coefficient a for the term with the highest power. It has the general form: \( ax^2 + bx + c = 0 \). To solve for x, one of the methods used is extracting square roots, which is ideal when the equation can be expressed in the form \( x^2 = k \) for some positive number k. Here's how it works:

  • Isolate the x-squared term on one side of the equation, so it stands alone.
  • Take the square root of both sides of the equation. Remember that the square root of x-squared is x.
  • Don't forget to consider both the positive and negative square roots of the number.

This technique is useful for its simplicity and is especially handier when dealing with equations not requiring more complex methods like completing the square or using the quadratic formula.
Square Root Simplification
The square root simplification process is key when extracting roots to solve quadratic equations. Simplifying a square root involves finding the prime factors of the number and pairing the factors, since the square root of a number n is a number which when multiplied by itself gives n. Consider breaking down the number inside the square root into its prime factors and use the property that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). For example:

  • Start with \(\sqrt{32}\).
  • Factor 32 into \(16 \times 2\), since 16 is a perfect square and its square root is easily known as 4.
  • Apply the property to get \(4\sqrt{2}\), as \(\sqrt{16} = 4\).

Through this process, we simplify expressions to their least radical form, making them easier to work with in equations and other mathematical operations.
Positive and Negative Roots
An essential point to remember when solving quadratic equations by extracting square roots is to consider both positive and negative roots. This is because a squared number can originate from both a positive and a negative number since both, when squared, will give a positive result. For example, \( x^2 = 32 \) will have solutions \( x = 4\sqrt{2} \) and \( x = -4\sqrt{2} \). Here's why both roots are significant:

  • A negative times a negative equals a positive; thus, the square of a negative number results in a positive number.
  • Failing to include both roots can lead to missing half of the possible solutions.
  • When graphing, this ensures both the intersection points of the parabola with the x-axis are accounted for.

Always including the positive and negative roots ensures a complete solution set for quadratic equations.

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

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. Proof Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\). (b) Use the function in part (a) to complete the table. What can you conclude? \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline\(n\) & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline\(R(n)\) & & & & & & & \\ \hline \end{tabular}

In Exercises 69-74, use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ f^{-1}\right)(-3) $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{2}, \quad x \geq 0 $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{3}+1 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

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.