/*! 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 62 Decide whether the given number ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 62

Decide whether the given number is a solution of the equation. \(2 x^{2}+1=19 ; 3\)

Short Answer

Expert verified
Yes, 3 is a solution to the equation.

Step by step solution

01

Substitute the Given Number

Substitute the given number, 3, into the equation to see if it satisfies it. The given equation is: \[ 2x^2 + 1 = 19 \]Replace x with 3: \[ 2(3)^2 + 1 \]
02

Simplify the Expression

Simplify the expression: \[ 2(3)^2 + 1 \]First, calculate the square of 3: \[ 3^2 = 9 \]Next, multiply by 2: \[ 2 \times 9 = 18 \]Finally, add 1: \[ 18 + 1 = 19 \]
03

Compare to Given Equation

Compare the simplified expression to the right-hand side of the given equation. Both are equal to 19: \[ 19 = 19 \]Since both sides of the equation match, the number 3 is a solution to the 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.

Substitution Method
The substitution method is a powerful and straightforward approach to solving equations. It involves replacing a variable with a given number to check if it satisfies the equation. In the problem, we substituted the number 3 into the variable x.
Simply follow these steps:
  • Identify the variable in the equation.
  • Replace the variable with the given number.
  • Simplify the expression to see if both sides are equal.
For example, in the equation \[ 2x^2 + 1 = 19 \], substituting x with 3 gives us the expression \[ 2(3)^2 + 1 \]. Then, we simplify to determine if the left side of the equation equals the right side (19).
By following these steps, you can easily check whether a given number is a solution for the equation.
Quadratic Equations
Quadratic equations are polynomials of the second degree, typically in the form of \[ ax^2 + bx + c = 0 \]. These types of equations can have up to two real solutions. The equation in the exercise, \[ 2x^2 + 1 = 19 \], fits into this category.
Quadratic equations can be solved using various methods, including:
  • Factoring
  • Completing the square
  • Quadratic formula
  • Substitution method (as shown in this problem)
Understanding how to identify and work with quadratic equations helps in solving a wide range of algebraic problems. By mastering these techniques, you can handle more complex equations with ease.
Simplifying Expressions
Simplifying expressions is a crucial step in solving algebraic problems. It involves performing operations like addition, subtraction, multiplication, and division to break down complex expressions into simpler forms. Here's how we simplified the expression from the exercise:
  • First, substitute the variable with the given number: \[ 2(3)^2 + 1 \]
  • Calculate the square of 3: \[ 3^2 = 9 \]
  • Multiply by 2: \[ 2 \times 9 = 18 \]
  • Add 1: \[ 18 + 1 = 19 \]
By breaking down each step, we simplified the original equation to check if the left side equals the right side. Simplifying expressions makes algebraic problems more manageable and helps you see the relationships between different parts of the equation clearly.

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 each expression for \(x=6, y=-4,\) and \(a=3\) $$ (-5+x)(-3+y)(3-a) $$

Write a numerical expression for each phrase and simplify. Three-fourths of the sum of \(-8\) and 12

Perform each indicated operation. $$ \frac{-5(-6)}{9-(-1)} $$

Write a numerical expression for each phrase and simplify. The product of \(-\frac{2}{3}\) and \(-\frac{1}{5},\) divided by \(\frac{1}{7}\)

Translate each phrase into a mathematical expression. Use \(x\) as the variable. Combine like terms when possible. Five times a number, added to the sum of the number and three
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.