/*! 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 48 Solve equation by using the squa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 48

Solve equation by using the square root property. Simplify all radicals. \((3-2 x)^{2}=70\)

Short Answer

Expert verified
x = \frac{3 - \sqrt{70}}{2} or x = \frac{3 + \sqrt{70}}{2}

Step by step solution

01

- Isolate the squared term

The given equation is \( (3-2x)^2 = 70 \). There is no need to isolate the squared term as it is already isolated.
02

- Take the square root of both sides

Take the square root of both sides of the equation to remove the square. \[ \sqrt{(3-2x)^2} = \sqrt{70} \] Simplify to get: \[ 3-2x = \pm \sqrt{70} \] Remember to consider both the positive and negative roots.
03

- Solve for x

We now have two separate linear equations to solve for \( x \): \[ 3-2x = \sqrt{70} \] and \[ 3-2x = -\sqrt{70} \] Solve each equation separately. For \( 3-2x = \sqrt{70} \): \[ -2x = \sqrt{70} - 3 \] \[ x = \frac{3 - \sqrt{70}}{2} \] For \( 3-2x = -\sqrt{70} \): \[ -2x = -\sqrt{70} - 3 \] \[ x = \frac{3 + \sqrt{70}}{2} \]

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 equations
In mathematics, solving equations involves finding the values of the variables that make the equation true. In our given example, \((3-2x)^2=70\), we need to determine the value of \(x\). When solving equations with square terms, we often use the Square Root Property. This involves taking the square root of both sides of the equation. There can be multiple solutions since both positive and negative roots need to be considered. The main steps include isolating the squared term (if not already isolated), taking the square root on both sides to remove the square, and then solving the resulting linear equations for the variable.
isolate the variable
Isolating the variable is a crucial step in solving equations, especially when dealing with squares. In the provided equation \((3-2x)^2=70\), we can see that the squared term is already isolated, meaning it stands alone on one side of the equation. If it wasn't, we would need to manipulate the equation to achieve isolation. This usually involves adding, subtracting, multiplying, or dividing both sides of the equation to ensure the squared term is by itself. The ultimate goal is to position the variable term in a way that makes it easier to solve in the subsequent steps.
radical simplification
Once the squared term is isolated, the next step is to simplify the radical. By taking the square root of both sides of the equation \((3-2x)^2 = 70\), we simplify it to: \sqrt{(3-2x)^2} = \sqrt{70}\. This simplifies further to \3-2x = \pm \sqrt{70}\, acknowledging that square roots have both positive and negative solutions. Simplifying radicals often means expressing the roots in their simplest form, and it鈥檚 essential when solving the resulting linear equations. Radical simplification helps reduce complex expressions into more manageable forms.
square roots
Square roots are fundamental in solving equations involving squared terms. In the problem, taking the square root of both sides \((3-2x)^2 = 70\) yields \3-2x = \pm \sqrt{70}\. Remember to consider both the positive and negative solutions since a square root has two possible values. To find the solutions for \(x\), solve the two separate equations: \3-2x = \sqrt{70}\ and \3-2x = -\sqrt{70}\. For the first equation: \3-2x = \sqrt{70}\ leads to \-2x = \sqrt{70}-3\ and \x = \frac{3-\sqrt{70}}{2}\. For the second equation: \3-2x = -\sqrt{70}\ results in \-2x = -\sqrt{70}-3\, so \x = \frac{3+\sqrt{70}}{2}\. Thus highlighting how the Square Root Property helps solve such 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

Solve equation by using the square root property. Simplify all radicals. \((2 x-5)^{2}-180=0\)

Two painters are painting a house in a development of new homes. One of the painters takes 2 hr longer to paint the house working alone than the other painter takes. When they do the job together, they can complete it in 4.8 hr. How long would it take the faster painter alone to paint the house? (Give your answer to the nearest tenth.)

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ p^{2}-3 p+4=0 $$

Simplify all radicals, and combine like terms. Express fractions in lowest terms. \(\frac{12-\sqrt{27}}{9}\)

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ 2 x^{2}+x+3=0 $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure 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.