/*! 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 39 Solve each equation by completin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 39

Solve each equation by completing the square $$ x^{2}-\frac{1}{2} x-\frac{3}{16}=0 $$

Short Answer

Expert verified
The solutions are \(x = \frac{3}{4}\) and \(x = -\frac{1}{4}\).

Step by step solution

01

Move the constant term to the other side

Start by moving the constant term from the left side to the right side. \[ x^{2} - \frac{1}{2} x = \frac{3}{16} \]
02

Complete the square

To complete the square, take half of the coefficient of the linear term, square it, and add it to both sides of the equation. The coefficient of the linear term here is \(-\frac{1}{2}\). Half of this coefficient is \(-\frac{1}{4}\), and its square is \(\left( -\frac{1}{4} \right)^{2} = \frac{1}{16}\).Add \(\frac{1}{16}\) to both sides of the equation:\[ x^{2} - \frac{1}{2}x + \frac{1}{16} = \frac{3}{16} + \frac{1}{16} \]This simplifies to:\[ x^{2} - \frac{1}{2}x + \frac{1}{16} = \frac{4}{16} \]\[ x^{2} - \frac{1}{2}x + \frac{1}{16} = \frac{1}{4} \]
03

Write the left side as a perfect square

The left side of the equation is now a perfect square trinomial. It can be written as the square of a binomial:\[ \left( x - \frac{1}{4} \right)^{2} = \frac{1}{4} \]
04

Solve for \(x\)

Take the square root of both sides of the equation:\[ \sqrt{\left( x - \frac{1}{4} \right)^{2}} = \pm \sqrt{\frac{1}{4}} \]\[ x - \frac{1}{4} = \pm \frac{1}{2} \]This results in two possible solutions for \(x\):\[ x - \frac{1}{4} = \frac{1}{2} \implies x = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \]\[ x - \frac{1}{4} = -\frac{1}{2} \implies x = -\frac{1}{2} + \frac{1}{4} = -\frac{1}{4} \]

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.

Solving Quadratic Equations
Solving quadratic equations is a fundamental skill in algebra. Quadratic equations have the form: ewline x^{2} + bx + c = 0. One efficient way to solve these is by completing the square. This method involves transforming the quadratic into a perfect square trinomial. Once we have the perfect square trinomial, it simplifies solving the quadratic equation significantly. Start by moving the constant to the right side and then complete the square on the left side.
Perfect Square Trinomial
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. For example, ewline x^{2} - 2x + 1 = (x - 1)^{2}. To convert a quadratic equation into a perfect square trinomial, follow these steps:

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

In going from Chicago to Atlanta, a car averages 45 miles per hour, and in going from Atlanta to Miami, it averages 55 miles per hour. If Atlanta is halfway between Chicago and Miami, what is the average speed from Chicago to Miami? Discuss an intuitive solution. Write a paragraph defending your intuitive solution. Then solve the problem algebraically. Is your intuitive solution the same as the algebraic one? If not, find the flaw.

Living at Home A study found that Millennials are more likely to live at home than GenXers at the same age. The model \(P=0.003 x^{2}-0.034 x+8.086\) gives the approximate percentage of 25 -to-35-year-olds living in their parents' home, \(x\) years after \(1960 .\) Use the model to estimate when the percentage of 25 -to-35-year-olds living in their parents' home was \(10 \% .\) Round to the nearest year. Data: Pew Research Center

Find the real solutions, if any, of each equation. Use any method. $$ x^{2}+\sqrt{2} x=\frac{1}{2} $$

Find the real solutions, if any, of each equation. Use any method. $$ 16 x^{2}-8 x+1=0 $$

Find the real solutions, if any, of each equation. Use any method. $$ x^{2}+x=4 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete 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.