/*! 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 37 Solve each equation by completin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 37

Solve each equation by completing the square $$ x^{2}+4 x=21 $$

Short Answer

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

Step by step solution

01

- Move constant to the right

Start by moving the constant term to the right side of the equation. Thus, rewrite the equation as:equation: \( x^2 + 4x = 21 \)
02

- Find the value to complete the square

Take half of the coefficient of \( x \), square it, and add it to both sides of the equation. The coefficient of \( x \) is 4. Half of 4 is 2, and 2 squared is 4. Add 4 to both sides:equation: \( x^2 + 4x + 4 = 21 + 4 \)
03

- Write as a square of a binomial

The left side of the equation can now be written as a perfect square binomial. Thus, the equation becomes:equation: \( (x + 2)^2 = 25 \)
04

- Solve for \( x \)

Next, take the square root of both sides of the equation to solve for \( x \). Don't forget to consider both the positive and negative roots:equation: \( x + 2 = \pm \sqrt{25} \)equation: \( x + 2 = \pm 5 \)Solving for \( x \):equation: \( x = -2 + 5 \) or \( x = -2 - 5 \)Thus, \( x = 3 \) or \( x = -7 \)

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.

quadratic equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). These types are special because they involve an \(x^2\) term, making them more complex than linear equations. In the given problem, \(x^2 + 4x = 21\), it is a quadratic equation since it contains the \(x^2\) term. To solve it, we'll use the method of completing the square, which transforms the quadratic into a perfect square trinomial.
solving equations
Solving equations is the process of finding the value of the variable that makes the equation true. For quadratic equations, there are several methods such as factoring, using the quadratic formula, and completing the square. Here, we focus on completing the square. The steps involve:
  • Moving the constant term to the other side
  • Finding a value to make the left side a perfect square trinomial
  • Converting it into a binomial square form
  • Solving by taking square roots
These steps systematically simplify the equation, making it easier to solve for the variable.
square roots
Square roots are essential in solving quadratic equations when using the completing the square method. The square root function undoes the squaring of a number. For instance, if we have \((x + 2)^2 = 25\), taking the square root of both sides gives us \(x + 2 = \pm \sqrt{25}\), which simplifies to \(x + 2 = \pm 5\). We must consider both positive and negative roots because squaring either a positive or a negative number yields the same result.
binomials
Binomials are algebraic expressions containing two terms. In our solution, we use the binomial \((x + 2)^2\) which emerges from completing the square. This transformation is crucial because it converts a quadratic equation into a more manageable form. Recognizing and manipulating binomials effectively allows us to simplify and solve complicated equations smoothly. In our case, \(x^2 + 4x + 4\) was rewritten as \((x + 2)^2\), making it simpler to handle.

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

IQ Tests A standard intelligence test has an average score of \(100 .\) According to statistical theory, of the people who take the test, the \(2.5 \%\) with the highest scores will have scores of more than \(1.96 \sigma\) above the average, where \(\sigma\) (sigma, a number called the standard deviation) depends on the nature of the test. If \(\sigma=12\) for this test and there is (in principle) no upper limit to the score possible on the test, write the interval of possible test scores of the people in the top \(2.5 \%\)

Describe three ways that you might solve a quadratic equation. State your preferred method; explain why you chose it.

Mixing Water and Antifreeze The cooling system of a certain foreign-made car has a capacity of 15 liters. If the system is filled with a mixture that is \(40 \%\) antifreeze, how much of this mixture should be drained and replaced by pure antifreeze so that the system is filled with a solution that is \(60 \%\) antifreeze?

Challenge Problem Solve: \(x-4<2 x-3 \leq \frac{x+5}{3}\)

A bathroom tub will fill in 15 minutes with both faucets open and the stopper in place. With both faucets closed and the stopper removed, the tub will empty in 20 minutes. How long will it take for the tub to fill if both faucets are open and the stopper is removed?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.