/*! 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 73 Complete the square to make a pe... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 73

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared. (a) \(p^{2}-22 p\) (b) \(y^{2}+5 y\) (c) \(m^{2}+\frac{2}{5} m\)

Short Answer

Expert verified
(a) \((p - 11)^{2}\), (b) \((y + \frac{5}{2})^{2}\), (c) \((m + \frac{1}{5})^{2}\)

Step by step solution

01

Understand the process

To complete the square, we need to transform a quadratic expression into a perfect square trinomial. This involves adding and subtracting the square of half the coefficient of the linear term.
02

Complete the square for part (a)

Given the expression is: \(p^{2} - 22p\). First, take the coefficient of the linear term, which is -22, divide it by 2 to get -11, and then square it to get 121. So, \(p^{2} - 22p + 121 - 121 = (p - 11)^{2} - 121\). The perfect square trinomial is \(p^{2} - 22p + 121 = (p - 11)^{2}\).
03

Complete the square for part (b)

Given the expression is: \(y^{2} + 5y\). First, take the coefficient of the linear term, which is 5, divide it by 2 to get \(\frac{5}{2}\), and then square it to get \(\frac{25}{4}\). So, \(y^{2} + 5y + \frac{25}{4} - \frac{25}{4} = (y + \frac{5}{2})^{2} - \frac{25}{4}\). The perfect square trinomial is \(y^{2} + 5y + \frac{25}{4} = (y + \frac{5}{2})^{2}\).
04

Complete the square for part (c)

Given the expression is: \(m^{2} + \frac{2}{5}m\). First, take the coefficient of the linear term, which is \(\frac{2}{5}\), divide it by 2 to get \(\frac{1}{5}\), and then square it to get \(\frac{1}{25}\). So, \(m^{2} + \frac{2}{5}m + \frac{1}{25} - \frac{1}{25} = (m + \frac{1}{5})^{2} - \frac{1}{25}\). The perfect square trinomial is \(m^{2} + \frac{2}{5}m + \frac{1}{25} = (m + \frac{1}{5})^{2}\).
05

Write the results as binomial squares

Using the results from the previous steps, write each perfect square trinomial as a binomial squared.For (a): \((p - 11)^{2}\)For (b): \((y + \frac{5}{2})^{2}\)For (c): \((m + \frac{1}{5})^{2}\)

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 expressions
A quadratic expression is a polynomial of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In the given exercises, we have quadratic expressions with no constant term \(c\). For instance, \(p^2 - 22p\) and \(y^2 + 5y\) are quadratic expressions because they contain a term with \(x^2\) and a linear term with \(x\). Understanding quadratic expressions is crucial because many algebraic methods, such as completing the square, rely on manipulating these expressions to reveal deeper properties.
perfect square trinomials
A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. This means it looks like \(a^2 + 2ab + b^2 = (a + b)^2\). To transform a quadratic expression into a perfect square trinomial, follow these steps:
  • Identify the quadratic expression, for example, \(y^2 + 5y\)
  • Take the coefficient of the linear term (\(b\)), which is 5, divide it by 2 to get \(\frac{5}{2}\)
  • Square the result to get \(\frac{25}{4}\)
By adding and subtracting this squared value, the expression becomes a perfect square trinomial: \(y^2 + 5y + \frac{25}{4} - \frac{25}{4} = (y + \frac{5}{2})^2 - \frac{25}{4}\). Now we have a perfect square trinomial simply represented as \(y^2 + 5y + \frac{25}{4} = (y + \frac{5}{2})^2\). This transformation helps to simplify solving or further manipulating the quadratic expression.
binomial squared
After completing the square and achieving a perfect square trinomial, the next step is writing the expression as a binomial squared. A binomial squared is a quadratic expression written as \((a + b)^2\) or \((a - b)^2\).
  • In exercise (a), from \(p^2 - 22p + 121\) we get \((p - 11)^2\).
  • In exercise (b), from \(y^2 + 5y + \frac{25}{4}\) we get \((y + \frac{5}{2})^2\).
  • In exercise (c), from \(m^2 + \frac{2}{5}m + \frac{1}{25}\) we get \((m + \frac{1}{5})^2\).
Writing the result as a binomial squared makes it easier to solve quadratic equations. It facilitates finding the roots or solutions of the equation by taking the square root of both sides. This method provides a neat and efficient way to handle quadratic equations, converting them into solvable forms with simplified expressions.

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

Graph each function using transformations. $$f(x)=(x-1)^{2}+5$$

Solve each inequality algebraically and write any solution in interval notation. $$-2 x^{2}+7 x+4 \geq 0$$

(a) solve graphically and (b) write the solution in interval notation. $$-x^{2}-3 x+18 \leq 0$$

Solve. Round answers to the nearest tenth. A ball is thrown vertically upward from the ground with an initial velocity of \(122 \mathrm{ft} / \mathrm{sec}\). Use the quadratic function \(h(t)=-16 t^{2}+122 t+0\) to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

Solve. Round answers to the nearest tenth. A cell phone company estimates that by charging \(x\) dollars each for a certain cell phone, they can sell \(8-\) \(x\) cell phones per day. Use the quadratic function \(R(x)=\) \(-x^{2}+8 x\) to find the revenue received per day when the selling price of a cell phone is \(x .\) Find the selling price that will give them the maximum revenue per day, and then find the amount of the maximum revenue.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.