/*! 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 58 Solve each equation by completin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 58

Solve each equation by completing the square. \(t^{2}+2 t-1=0\)

Short Answer

Expert verified
t = -1 +\sqrt{2} or t = -1 - \sqrt{2}

Step by step solution

01

Move the constant term to the other side

Start by moving the constant term (-1) to the other side of the equation. So, rewrite the equation as: \[t^2 + 2t = 1\]
02

Add the square of half the coefficient of t

Take the coefficient of t, which is 2, and divide by 2 and then square it. \[ \left( \frac{2}{2} \right)^2 = 1\] Add 1 to both sides of the equation so it becomes: \[t^2 + 2t + 1 = 1 + 1\] which simplifies to: \[t^2 + 2t + 1 = 2\]
03

Write the left side as a squared binomial

The left side of the equation \( t^2 + 2t + 1 \) is a perfect square trinomial and can be factored as: \[ (t + 1)^2 = 2\]
04

Solve the equation

Take the square root of both sides to solve for t: \[ t + 1 = \pm \sqrt{2}\] So, we have 't' as: \[ t = -1 \pm \sqrt{2}\]

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.

algebraic equations
Algebraic equations involve finding the value of unknown variables that satisfy the equation. They are usually written as expressions set equal to values or other expressions. For example, in the equation \(t^2 + 2t - 1 = 0\), \(t\) is the variable we need to solve for.
To solve algebraic equations, we often use various methods such as substitution, elimination, and for quadratic equations, completing the square.
The goal is to isolate the variable on one side of the equation. Understanding these concepts is crucial for solving both simple and complex equations.
quadratic equations
Quadratic equations are algebraic equations where the highest power of the variable is 2. They have the general form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants.
In our example, \(t^2 + 2t - 1 = 0\), the equation is already in quadratic form with \(a = 1\), \(b = 2\), and \(c = -1\).
Quadratic equations can be solved by several methods:
  • Factoring
  • Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
  • Completing the square
Our solution uses completing the square, which transforms the quadratic into a binomial square.
solving by square root
The solving by square root method comes into play after we have rewritten the quadratic equation as a perfect square trinomial. In our example, after completing the square, we had:
\( (t + 1)^2 = 2 \)
This is now a binomial square, and we solve by taking the square root of both sides:
\( t + 1 = \pm \sqrt{2} \)
By isolating \(t\), we subtract 1 from both sides, yielding:
\( t = -1 \pm \sqrt{2} \)
This technique simplifies the process of solving quadratics, especially when typical factoring is challenging.
It highlights how transforming the equation format can make finding solutions straightforward.

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

Two pipes together can fill a tank in \(2 \mathrm{hr}\). One of the pipes, used alone, takes \(3 \mathrm{hr}\) longer than the other to fill the tank. How long would each pipe take to fill the tank alone?

Solve each equation. Check the solutions. $$3-2(x-1)^{-1}=(x-1)^{-2}$$

Solve each equation. Check the solutions. $$x^{4}-37 x^{2}+36=0$$

Vera flew for \(6 \mathrm{hr}\) at a constant rate. She traveled \(810 \mathrm{mi}\) with the wind, then turned around and traveled 720 mi against the wind. The wind speed was a constant 15 mph. Find the rate of the plane. $$\begin{array}{|l|c|c|c|}\hline & d & r & t \\\\\hline \text { With Wind } & 810 & & \\\\\text { Against Wind } & 720 & & \\\\\hline\end{array}$$

Solve each equation. Check the solutions. $$4 t=\sqrt{8 t+3}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.