/*! 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 34 Find all complex-number solution... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 34

Find all complex-number solutions. Let \(g(x)=(x-2)^{2} .\) Find \(x\) such that \(g(x)=25\)

Short Answer

Expert verified
The solutions are \(x = 7\) and \(x = -3\).

Step by step solution

01

- Set the equation equal to 25

We are given the function \(g(x) = (x - 2)^2\) and asked to find \(x\) for which \(g(x) = 25\). Setting the function equal to 25, we get: \[ (x - 2)^2 = 25 \]
02

- Take the square root of both sides

To solve for \(x - 2\), take the square root of both sides: \[ \sqrt{(x - 2)^2} = \sqrt{25} \] This simplifies to two possible equations: \[ x - 2 = 5 \] and \[ x - 2 = -5 \]
03

- Solve for x

Solve each equation separately: For \( x - 2 = 5 \): \[ x = 5 + 2 \] which simplifies to \[ x = 7 \] For \( x - 2 = -5 \): \[ x = -5 + 2 \] which simplifies to \[ x = -3 \]

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
Quadratic equations are polynomial equations of degree 2, generally in the form of \[ ax^2 + bx + c = 0 \]. To solve these equations, we have several methods, but one of the most straightforward techniques is using the **Square Root Property**. This property is particularly useful when the equation can be rearranged as \[ (x - h)^2 = k \].
If we follow the given exercise,\[ g(x) = (x - 2)^2 = 25 \], we can directly apply this method.
  • First, make sure the quadratic is in the form that allows us to use the Square Root Property efficiently.
  • Then, by setting \[ (x - h)^2 = k \], solve by taking the square root, which gives us two possible solutions.
These solutions represent the points where the original quadratic equals 25. Let's break this down further with the Square Root Property.
Square Root Property
The Square Root Property states that if \[ (x - h)^2 = k \], then \[ x - h = \pm \sqrt{k} \]. This property is incredibly useful in simplifying the process of solving quadratic equations.
For the given example of \[ (x - 2)^2 = 25 \], here鈥檚 how we use the property:
  • Recognize that we can take the square root of both sides of the equation: \[ \sqrt{(x - 2)^2} = \sqrt{25} \]
  • This simplifies to two cases: \[ x - 2 = 5 \] and \[ x - 2 = -5 \]
  • From these, we can solve for \[ x \] by adding 2 back to each equation.
This directly leads us to our two possible solutions:
  • If \[ x - 2 = 5 \], then \[ x = 7 \]
  • If \[ x - 2 = -5 \], then \[ x = -3 \]
This property allows for a quick and effective solution without tedious factoring or the quadratic formula.
Finding Complex-Number Solutions
When dealing with quadratic equations, sometimes the solutions aren't always real numbers. If we squared a negative number, we'd end up with a negative inside the square root, indicating complex solutions.
Complex numbers have the form \[ a + bi \], where \[ i \] is the imaginary unit and \[ i^2 = -1 \].
In our current exercise, both solutions are real (\[ x = 7 \] and \[ x = -3 \]). However, if instead of \[ (x - 2)^2 = 25 \], we had \[ (x - 2)^2 = -25 \], our approach would be different:
  • Using the Square Root Property again, \[ (x - 2)^2 = -25 \] becomes \[ x - 2 = \pm \sqrt{-25} \].
  • Recognize that \[ \sqrt{-25} = 5i \], leading to \[ x - 2 = \pm 5i \].
  • This means \[ x = 2 + 5i \] and \[ x = 2 - 5i \].
These solutions include imaginary parts, illustrating why they are called complex numbers. Understanding these concepts provides greater depth in solving any quadratic equation.

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

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (4,-7) $$

Working together, Tanner and Joel can grade their students' projects in 2 hr. Working alone, it would take Tanner 2 hr longer than it would take Joel to grade the projects. How long would it take Joel, working alone, to grade the projects?

Solve. Stan's Subaru travels 280 mi averaging a certain speed. If the car had gone 5 mph faster, the trip would have taken 1 hr less. Find Stan's average speed.

Total Profit. Derex, Inc., determines that its total profit function is given by $$P(x)=-3 x^{2}+630 x-6000.$$ a) Find all values of \(x\) for which Derex makes a profit. b) Find all values of \(x\) for which Derex loses money.

Find each solution set. $$\left|\frac{x+2}{x-1}\right| \leq 3$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision 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.