/*! 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 35 Find all real solutions of the e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 35

Find all real solutions of the equation. $$ x^{2}+3 x+1=0 $$

Short Answer

Expert verified
The real solutions are \( x = \frac{-3 + \sqrt{5}}{2} \) and \( x = \frac{-3 - \sqrt{5}}{2} \).

Step by step solution

01

Identify the Type of Equation

The given equation is a quadratic equation: \( x^2 + 3x + 1 = 0 \). We need to find its roots using the quadratic formula, since it may not factorize easily.
02

Recall the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), used for solving equations of the form \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = 3 \), and \( c = 1 \).
03

Calculate the Discriminant

Calculate the discriminant using \( b^2 - 4ac \). Substituting the known values, we have: \( 3^2 - 4 \times 1 \times 1 = 9 - 4 = 5 \). Since the discriminant is positive, there are two distinct real roots.
04

Apply the Quadratic Formula

Substitute \( a \), \( b \), and \( c \) into the quadratic formula: \( x = \frac{-3 \pm \sqrt{5}}{2} \).
05

Simplify the Expression

Compute the two possible solutions: \( x = \frac{-3 + \sqrt{5}}{2} \) and \( x = \frac{-3 - \sqrt{5}}{2} \). These are the real solutions to the equation.

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.

Understanding the Quadratic Formula
Quadratic equations like the one given, \(x^2 + 3x + 1 = 0\), appear often in algebra. The quadratic formula is a robust tool for finding the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). When you have coefficients \(a\), \(b\), and \(c\), you can find the solutions for \(x\) using:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
This formula provides a straightforward path to the roots, whether or not the equation can be factored easily. Here, the coefficients are \(a = 1\), \(b = 3\), and \(c = 1\). The formula accounts for every possibility, from equations with two distinct roots to cases where a solution might repeat.
The Role of the Discriminant
The discriminant is the part of the quadratic formula under the square root: \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of a quadratic equation. Here's how:
  • If the discriminant is positive, as in this case \(3^2 - 4 \times 1 \times 1 = 5\), it indicates that the quadratic equation has two distinct real solutions.
  • If the discriminant equals zero, there is exactly one real solution, and the root is repeated.
  • If the discriminant is negative, the quadratic equation has no real solutions; instead, you'll get two complex solutions.
Thus, in the process of solving \(x^2 + 3x + 1 = 0\), the positive discriminant \(5\) means we will find two distinct real solutions.
Finding Real Solutions
Real solutions to a quadratic equation can be explicitly calculated using the quadratic formula once the discriminant has been evaluated. As seen, the given equation is solved as follows:
Plug in \(a = 1\), \(b = 3\), and the discriminant \(\sqrt{5}\) into the quadratic formula:
  • \(x = \frac{-3 \pm \sqrt{5}}{2}\)
This yields the two separate solutions:
  • \(x = \frac{-3 + \sqrt{5}}{2}\)
  • \(x = \frac{-3 - \sqrt{5}}{2}\)
These solutions are called real because they do not include imaginary numbers and can be plotted on a real number line. Therefore, whenever you encounter a quadratic equation, applying the quadratic formula and evaluating the discriminant will guide you to finding its real solutions effectively.

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

Evaluate the expression and write the result in the form a bi. $$ (3-4 i)(5-12 i) $$

Evaluate the expression and write the result in the form a bi. $$ 6 i-(4-i) $$

Relationship Between Roots and Coefficients The Quadratic Formula gives us the roots of a quadratic equation from its coefficients. We can also obtain the coefficients from the roots. For example, find the roots of the equation \(x^{2}-9 x+20=0\) and show that the product of the roots is the constant term 20 and the sum of the roots is 9 , the negative of the coefficient of \(x\) . Show that the same relationship between roots and coefficients holds for the following equations: $$ \begin{array}{l}{x^{2}-2 x-8=0} \\ {x^{2}+4 x+2=0}\end{array} $$ Use the Quadratic Formula to prove that in general, if the equation \(x^{2}+b x+c=0\) has roots \(r_{1}\) and \(r_{2},\) then \(c=r_{1} r_{2}\) and \(b=-\left(r_{1}+r_{2}\right)\)

Evaluate the expression and write the result in the form a bi. $$ i^{3} $$

Find the real and imaginary parts of the complex number. $$ \frac{-2-5 i}{3} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.