/*! 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 27 \(3 x^{2}+2 x-2=0\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 27

\(3 x^{2}+2 x-2=0\)

Short Answer

Expert verified
The solutions are \(x = \frac{-1 \pm \sqrt{7}}{3}\).

Step by step solution

01

Identify the Equation Type

The given equation is a quadratic equation in the form of \(ax^2 + bx + c = 0\). Here, \(a = 3\), \(b = 2\), and \(c = -2\).
02

Use the Quadratic Formula

To solve the quadratic equation, use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
03

Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). Substituting the values gives \(2^2 - 4 \times 3 \times (-2) = 4 + 24 = 28\).
04

Apply the Quadratic Formula

Substitute \(b = 2\), \(a = 3\), and the discriminant (28) into the quadratic formula. This gives \(x = \frac{-2 \pm \sqrt{28}}{6}\).
05

Simplify the Expression

Simplify \(\sqrt{28}\) to \(2\sqrt{7}\). The expression becomes \(x = \frac{-2 \pm 2\sqrt{7}}{6}\). Simplify further to \(x = \frac{-1 \pm \sqrt{7}}{3}\).
06

Find the Solutions

Thus, the solutions are \(x = \frac{-1 + \sqrt{7}}{3}\) and \(x = \frac{-1 - \sqrt{7}}{3}\).

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
The quadratic formula is a crucial tool for solving quadratic equations, which typically take the form \(ax^2 + bx + c = 0\). Using the coefficients \(a\), \(b\), and \(c\), we can apply the quadratic formula:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
This formula allows us to find the solutions of a quadratic equation, which are the values of \(x\) that satisfy the equation. To use the formula, simply plug in the values of \(a\), \(b\), and \(c\) from your equation. The symbol "\(\pm\)" indicates that there are generally two solutions, which consider both the positive and negative square roots.
Discriminant
In the context of the quadratic formula, the discriminant plays a key role. It is expressed as \(b^2 - 4ac\) and can tell us a lot about the nature of the solutions without having to compute them fully.
  • If the discriminant is positive, there are two distinct real solutions.
  • If it is zero, there is exactly one real solution (sometimes called a repeated or double root).
  • If it is negative, the solutions are complex or imaginary numbers.
For our specific equation of \(3x^2 + 2x - 2 = 0\), the discriminant calculates to 28:\[\begin{align*} 2^2 - 4 \cdot 3 \cdot (-2) & = 4 + 24 \ & = 28 \end{align*}\]Since 28 is positive, we have two distinct real solutions.
Solutions of Equations
After understanding the discriminant, we move onto finding the actual solutions using the quadratic formula. From the equation \(3x^2 + 2x - 2 = 0\), substitute into the formula:\[x = \frac{-2 \pm \sqrt{28}}{6}\]To simplify \(\sqrt{28}\), recognize it as \(2\sqrt{7}\), so the expression becomes:\[x = \frac{-2 \pm 2\sqrt{7}}{6}\]By simplifying further, divide each term in the numerator by 2:\[x = \frac{-1 \pm \sqrt{7}}{3}\]Thus, the final solutions for our quadratic equation are:
  • \(x = \frac{-1 + \sqrt{7}}{3}\)
  • \(x = \frac{-1 - \sqrt{7}}{3}\)
These solutions are the values that satisfy \(3x^2 + 2x - 2 = 0\). They are the points where the graph of this quadratic equation intersects the x-axis.

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

A class trip was to cost $$\$ 3000$$. If there had been ten more students, it would have cost each student $$\$ 25$$ less. How many students took the trip?

On a 50 -mile bicycle ride, Irene averaged 4 miles per hour faster for the first 36 miles than she did for the last 14 miles. The entire trip of 50 miles took 3 hours. Find her rate for the first 36 miles.

Use the quadratic formula to solve each of the following equations. Express the solutions to the nearest hundredth. $$x^{2}-5 x-19=0$$

Use the quadratic formula to solve each of the following quadratic equations. $$2 x^{2}+7 x-6=0$$

Use the quadratic formula to solve each of the following quadratic equations. $$3 t^{2}+6 t+5=0$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.