/*! 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 112 Solve each equation, rounding yo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 112

Solve each equation, rounding your answer to four significant digits where necessary. $$ x^{2}-(2-3 x)^{2}=0 $$

Short Answer

Expert verified
The solutions to the equation \(x^2 - (2-3x)^2 = 0\) are \(x \approx 0.7564\) and \(x \approx 0.7436\), rounded to four significant digits.

Step by step solution

01

Expand the squared binomial

First, we need to expand the squared binomial term: (2-3x)^2. Use the formula (a-b)^2 = a^2 - 2ab + b^2, where a = 2 and b = 3x. \[ (2-3x)^2 = 2^2 - 2(2)(3x) + (3x)^2 = 4 - 12x + 9x^2 \]
02

Rewrite the equation with the expanded term

Now rewrite the equation with the expanded term: \[ x^2 - (4 - 12x + 9x^2) = 0 \]
03

Simplify the equation

Combine the like terms and simplify the equation: \[ x^2 - 4 + 12x - 9x^2 = 0 \Rightarrow -8x^2 + 12x + x^2 - 4 = 0 \] Now, we can rewrite it in decreasing power order: \[ -8x^2 + x^2 + 12x - 4 = 0 \]
04

Solve the quadratic equation

After combining like terms, we get the quadratic equation: \[ -7x^2 + 12x - 4 = 0 \] Now, to solve the quadratic equation, -7x^2 + 12x - 4 = 0, we can either use the quadratic formula or factoring techniques. Since the coefficients are not friendly numbers, we will use the quadratic formula: For the equation ax^2 + bx + c = 0, the quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In our case, a = -7, b = 12, and c = -4.
05

Calculate the solution using the quadratic formula

Plug the values of a, b, and c into the quadratic formula and evaluate the expression: \[ x = \frac{-12 \pm \sqrt{12^2 - 4(-7)(-4)}}{2(-7)} \] Calculate the discriminant, b^2 - 4ac = 144 - 112 = 32. Now, we have: \[ x = \frac{-12 \pm \sqrt{32}}{-14} \] So, there are two possible solutions for x: \[ x_1 = \frac{-12 + \sqrt{32}}{-14} \approx 0.7564 \] and \[ x_2 = \frac{-12 - \sqrt{32}}{-14} \approx 0.7436 \] Thus, the rounded solutions to the equation x^2 - (2-3x)^2 = 0 are x 鈮� 0.7564 and x 鈮� 0.7436, to four significant digits.

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!

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

Convert each expression in Exercises into its technology formula equivalent as in the table in the text.\(\frac{4 \times 2}{\left(\frac{2}{3}\right)}\)

\(y^{4}+2 y^{2}-3\)

Convert each expression in Exercises into its technology formula equivalent as in the table in the text.\(3\left(\frac{1+4}{100}\right)^{-3}\)

By any method, determine all possible real solutions of each equation Check your answers by substitution. \(2 x^{2}-x-3=0\)

\(2 x^{2}-3 x-2\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.