/*! 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 98 Solve the quadratic equation by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 98

Solve the quadratic equation by the method of your choice. $$2 x^{2}+3 x=1$$

Short Answer

Expert verified
The solutions to the quadratic equation \(2x^{2}+3x-1=0\) are \(x=(-3+鈭�17)/4\) and \(x=(-3-鈭�17)/4\).

Step by step solution

01

Write the quadratic equation in standard form

First, write the given equation in standard form ax虏+bx+c=0. \n So, the equation 2x虏+3x-1=0 in standard form is 2x虏+3x-1=0.
02

Identify the coefficients.

In the equation 2x虏+3x-1=0, identify the values of a, b, and c. For this equation, a = 2, b = 3 and c = -1.
03

Substitute the values into the quadratic formula and simplify.

Substitute the values a, b, and c into the quadratic formula \(-b卤鈭�(b虏-4ac)/2a\). Then simplify the formula.\n Substituting the values gives: \(x= -b卤鈭�((3)虏-4*2(-1))/(2*2)\).\nThis simplifies to: \(x= -3卤鈭�(9+8)/4\), and further simplifies to \(x= -3卤鈭�17/4\).
04

Calculate the roots of the equation.

Find the two potential solutions, the roots, based on the equation \(x= -3卤鈭�17/4)\). This equation gives two solutions for x: \(x1=(-3+鈭�17)/4\) and \(x2=(-3-鈭�17)/4\).

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 Formula
When faced with a quadratic equation like the one in our problem, the quadratic formula is one of the most reliable methods to find its roots. The quadratic formula is given by \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). Here, \(a\), \(b\), and \(c\) are the coefficients of the terms in a quadratic equation in its standard form.
  • The quadratic formula allows you to find the solutions for any quadratic equation regardless of whether it can be factored easily or not.
  • "\(\pm\)" means there will be two solutions, sometimes called the roots of the equation.
The real beauty of the quadratic formula lies in its versatility, enabling you to solve equations without needing to arrange terms or guess possible factors.
Standard Form of a Quadratic Equation
The equation \(2x^2 + 3x = 1\) must first be transformed into its standard form. A quadratic equation's standard form is \(ax^2 + bx + c = 0\).
  • "\(a\)", "\(b\)", and "\(c\)" are the coefficients where \(a\) is the coefficient for \(x^2\), \(b\) for \(x\), and \(c\) is the constant term.
  • "\(0\)" explicitly shows that the equation is set to equal zero to facilitate solving.
For our equation, move 1 from the right side to the left side: \(2x^2 + 3x - 1 = 0\).This is now in standard form, making it easier to substitute into the quadratic formula.
Solving Quadratic Equations
Solving a quadratic equation often involves rearrangement or direct application of methods like the quadratic formula. Here鈥檚 a simplified breakdown:
  • Step 1: Check that the equation is in standard form.
  • Step 2: Identify the values for \(a\), \(b\), and \(c\).
  • Step 3: Substitute these values into relevant formulas, like the quadratic formula.
  • Step 4: Simplify the formula to find potential solutions, known as the roots.
For our equation \(2x^2 + 3x - 1 = 0\), \(a = 2\), \(b = 3\), and \(c = -1\). Using the quadratic formula yields two potential solutions, providing a clear pathway to solve even more complex quadratic equations.
Discriminant in Quadratics
The discriminant is a critical part of the quadratic formula, found under the square root symbol: \(b^2 - 4ac\). It provides key insights into the nature of the solutions:
  • If the discriminant \(b^2 - 4ac\) is greater than 0, there are two distinct real roots.
  • If it equals 0, there is exactly one real root, indicating the roots are identical.
  • If less than 0, the roots are complex and no real number solutions exist.
For the equation \(2x^2 + 3x - 1 = 0\), the discriminant is \((3)^2 - 4(2)(-1) = 9 + 8 = 17\), which is greater than 0. This confirms the existence of two distinct real roots, each solution providing valuable insights into the behavior of the quadratic equation.

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although \(20 x^{3}\) appears in both \(20 x^{3}+8 x^{2}\) and \(20 x^{3}+10 x\) I'll need to factor \(20 x^{3}\) in different ways to obtain each polynomial's factorization.

Solve each equation. $$7-7 x=(3 x+2)(x-1)$$

What is the discriminant and what information does it provide about a quadratic equation?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The cube root of \(-8\) is not a real number.

Find all integers \(b\) so that the trinomial can be factored. $$x^{2}+4 x+b$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Applied 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.