/*! 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 2 To complete the square of \(x^{2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 2

To complete the square of \(x^{2}-4 x,\) add _________.

Short Answer

Expert verified
4

Step by step solution

01

Identify the Coefficient of x

Start by identifying the coefficient of the linear term, which is the coefficient of the term with x. In this case, it is -4.
02

Divide the Coefficient by 2

Take the coefficient of x, divide it by 2, and then square the result. For the coefficient -4: \[ \frac{-4}{2} = -2 \]
03

Square the Result

Now, square the result obtained in the previous step. \[ (-2)^2 = 4 \]
04

Verify the Addition

The number to be added to complete the square is 4.

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
A quadratic equation is an equation of the form \[ ax^2 + bx + c = 0 \] where
    \t
  • \( a \) is the coefficient of the squared term \( x^2 \)
    • \t
  • \( b \) is the coefficient of the linear term \( x \)
    • \t
  • \( c \) is the constant term
Quadratic equations can have up to two solutions. These solutions are derived using various methods, such as factoring, using the quadratic formula, and completing the square.

In this exercise, we use the method of completing the square to transform our standard quadratic expression into a form that makes it easier to solve. This means we turn our equation into a perfect square trinomial. This will enable us to easily find the value of \( x \).
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying algebraic expressions to make them easier to work with or solve. In the context of completing the square, here鈥檚 how we proceed:

You start with the quadratic expression \( x^2 - 4x \).
    \t
  • Identify the coefficient of the linear term, which is -4.
    • \t
  • Divide this coefficient by 2.
    • \t
  • Square the result of the division.
By performing these algebraic steps: \[ \left( -\frac{4}{2} \right)^2 = 4 \] we will find the needed constant to complete our square. Adding 4 to \( x^2 - 4x \) transforms it into a perfect square trinomial.
coefficient
A coefficient is a number used to multiply a variable. In a quadratic equation like \( ax^2 + bx + c = 0 \), there are three coefficients:
    \t
  • \( a \), the coefficient of \( x^2 \)
    • \t
  • \( b \), the coefficient of \( x \)
    • \t
  • \( c \), the constant term
For the equation \( x^2 - 4x \), the coefficients are as follows:
    \t
  • \( a = 1 \) (implied as it鈥檚 not explicitly shown)
    • \t
  • \( b = -4 \) (the coefficient of \( x \) )
    • \t
  • \( c \) usually represents a constant but in this case, it鈥檚 zero
The process of completing the square involves manipulating the coefficients correctly. This improves our ability to solve the quadratic equation efficiently. In particular, working with the coefficient of the linear term鈥攄ividing it by 2 and squaring it鈥攁llows us to create a perfect square trinomial.

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

Bill's train leaves at 8: 06 an and accelerates at the rate of 2 meters per second per second. Bill, who can run 5 meters per second, arrives at the train station 5 seconds after the train has left and runs for the train. (a) Find parametric equations that model the motions of the train and Bill as a function of time. [Hint: The position \(s\) at time \(t\) of an object having acceleration \(a\) is \(s=\frac{1}{2} a t^{2}\). (b) Determine algebraically whether Bill will catch the train. If so, when? (c) Simulate the motion of the train and Bill by simultaneously graphing the equations found in part (a).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. \(M\) varies directly as \(x\) and the square of \(d .\) If \(M=90\) when \(x=14\) and \(d=3,\) find the proportionality constant.

A hall 100 feet in length is to be designed as a whispering gallery. If the foci are located 25 feet from the center, how high will the ceiling be at the center?

For \(f(x)=-\frac{1}{2} \sin (3 x+\pi)+5,\) find the amplitude, period, phase shift, and vertical shift.

Transform the polar equation \(r=6 \sin \theta\) to an equatiol in rectangular coordinates. Then identify and graph the equation.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.