/*! 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 28 Solve quadratic equation by comp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 28

Solve quadratic equation by completing the square. \(2 x^{2}-4 x-2=0\)

Short Answer

Expert verified
The solution to the quadratic equation \(2x^2-4x-2=0\) by completing the square is \(x=1\)

Step by step solution

01

Write the equation in standard form

First, it's best to rewrite the given equation in the standard form of quadratic equation, which is \(ax^2+bx+c=0\). Now divide each term by 2 to simplify, which gives \(x^2-2x-1=0\).
02

Complete the square

To complete the square, the equation is rearranged as \((x^2-2x)=-1\). Next, add the square of half of the coefficient of x, which is \((-2/2)^2=1\), to both sides of the equation. This gives \((x^2-2x+1)=-1+1=0\). The left hand side of the equation forms a perfect square and is written as \((x-1)^2=0\).
03

Solve for the variable x

Finally, solve for x by taking the square root of both sides of the equation. This gives \(x-1=0\) and thus \(x=1\)

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 simply equations of the second degree, typically in the form \(ax^2 + bx + c = 0\). Understanding the structure is crucial to solving them. Here are the basic terms used in these equations:
  • \(a\), \(b\), and \(c\) are constants, with \(a\) never being zero.
  • \(x\) represents an unknown variable that we strive to solve.
  • The highest power of \(x\) is 2, which is what makes it a quadratic equation.
Quadratic equations can represent various real-life situations, such as projectile motion. Solving them can provide insights into the minimum and maximum values and help estimate roots, intersections, and other critical points. The equation we are examining, \(2x^2 - 4x - 2 = 0\), fits this description and can be systematically solved using different methods.
Algebra
Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It provides a simple way to express complex relationships and solve equations, such as quadratic equations. Here are some algebraic principles crucial to solving our example problem:
  • Standard Form: The quadratic equation is rearranged to \(ax^2 + bx + c = 0\) to make it easier to handle.
  • Simplification: In our problem, the equation \(2x^2 - 4x - 2 = 0\) is simplified by dividing all terms by 2, resulting in \(x^2 - 2x - 1 = 0\). This step makes calculations more manageable.
  • Rearrangement: Completing the square requires rearranging terms to facilitate square formation, as in \((x^2 - 2x) = -1\).
By understanding these algebraic basics, we can systematically approach solving quadratic equations using various techniques, aiding in our mastery of algebra.
Solving Equations
Solving equations, especially quadratic ones like \(x^2 - 2x - 1 = 0\), involves specific techniques that transform the equation into a form where the solution can be readily obtained. For example, completing the square is a robust method:
  • Identifying Components: Recognize the components required to turn part of the equation into a perfect square, such as taking half of the coefficient of \(x\) (i.e., \(-2/2\)) and squaring it \((-1)^2 = 1\).
  • Forming a Perfect Square: Add this squared value to both sides, transforming the equation into a perfect square trinomial, \((x-1)^2 = 0\).
  • Finding the Solution: Taking the square root of both sides dissolves the square, leaving us with a linear equation \(x - 1 = 0\). This straightforwardly leads us to the solution \(x = 1\).
This method not only simplifies calculations but also provides a clear pathway from a complex equation to its solution. Mastery of these steps is essential for efficiently solving quadratic equations.

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

Describe how to find a parabola's vertex.

The weight of a human fetus is modeled by the formula \(W=3 t^{2}\) where \(W\) is the weight, in grams, and \(t\) is the time, in weeks, with \(0 \leq t \leq 39 .\) Use this formula to solve Exercises \(73-74\) After how many weeks does the fetus weigh 192 grams?

Solve: $$x^{2}+2 \sqrt{3} x-9=0$$

Find and simplify. \(\frac{f(x)-f(h)}{x-h}\) $$f(x)=8 x+9$$

Will help you prepare for the material covered in the next section. $$\text { Factor: } x^{2}+5 x+\frac{25}{4}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.