/*! 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 96 Solve. Round any irrational solu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 96

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}-5 x+1=0 $$

Short Answer

Expert verified
The solutions are approximately 4.791 and 0.209.

Step by step solution

01

Identify the coefficients

In the quadratic equation \(ax^2 + bx + c = 0\), identify the values of \(a\), \(b\), and \(c\) from the equation \(x^2 - 5x + 1 = 0\). Here, \(a = 1\), \(b = -5\), and \(c = 1\).
02

Use the quadratic formula

The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute the values \(a = 1\), \(b = -5\), and \(c = 1\) into the quadratic formula.
03

Calculate the discriminant

First, calculate the discriminant \(b^2 - 4ac\): \((-5)^2 - 4 \cdot 1 \cdot 1 = 25 - 4 = 21\).
04

Calculate the solutions

Substitute the discriminant back into the quadratic formula: \[x = \frac{5 \pm \sqrt{21}}{2}\]. This results in two solutions: \[x = \frac{5 + \sqrt{21}}{2}\]and \[x = \frac{5 - \sqrt{21}}{2}\].
05

Simplify and round the solutions

Calculate the numerical values using a calculator: \[x = \frac{5 + \sqrt{21}}{2} \approx 4.791\] and \[x = \frac{5 - \sqrt{21}}{2} \approx 0.209\]. Round both solutions to the nearest thousandth.

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
To solve quadratic equations, we often use a special formula known as the quadratic formula. This formula helps us find the roots (solutions) of any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. This formula might look complicated at first, but once you break it down step by step, it's simpler to use.
  • Firstly, make sure your equation is in the standard form \(ax^2 + bx + c = 0\).
  • Then, identify the coefficients \(a\), \(b\), and \(c\).
  • Substitute these values into the formula.
This formula works for all quadratic equations and provides two possible solutions due to the \(\pm\) symbol.
discriminant
The discriminant is part of the quadratic formula and plays a crucial role in determining the nature of the roots. It is expressed as \(b^2 - 4ac\).
  • If the discriminant is positive, there are two distinct real roots.
  • If it is zero, there is exactly one real root (also called a repeated root).
  • If it is negative, the roots are complex or imaginary.
For our example equation \(x^2 - 5x + 1 = 0\), we calculated the discriminant as \(21\) (since \((-5)^2 - 4 \cdot 1 \cdot 1 = 25 - 4 = 21\)). Since 21 is positive, we know that our quadratic equation has two distinct real roots.
solving quadratic equations
Solving quadratic equations involves a few steps using the quadratic formula. First, identify the coefficients \(a\), \(b\), and \(c\) in your equation. For the equation \(x^2 - 5x + 1 = 0\), we found \(a = 1\), \(b = -5\), and \(c = 1\).
Next, calculate the discriminant: \(b^2 - 4ac = 21\).
Substitute \(a\), \(b\), and the discriminant into the quadratic formula: \[x = \frac{5 \pm \sqrt{21}}{2}\].
Finally, solve for the two possible values of \(x\). These are:
  • \[x = \frac{5 + \sqrt{21}}{2} \approx 4.791\]
  • \[x = \frac{5 - \sqrt{21}}{2} \approx 0.209\]
Each step helps incrementally and ensures accuracy in finding the solutions.

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

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ p^{2} q-25 q+3 p^{2}-75 $$

Factor completely. Assume that variables in exponents represent positive integers. $$ x^{4 a}-49 y^{2 a} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ x^{2}+2 x y+y^{2}-9 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ q^{2}+1 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ (a+b)^{2}-9 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.