/*! 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 94 Evaluate \(\sqrt{b^{2}-4 a c}\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 94

Evaluate \(\sqrt{b^{2}-4 a c}\) for each set of values. See Section 7.3. $$ a=1, b=6, c=2 $$

Short Answer

Expert verified
The value is \(2\sqrt{7}\).

Step by step solution

01

Identify the Given Values

First, identify the values given for \(a\), \(b\), and \(c\). In this problem, we have \(a=1\), \(b=6\), and \(c=2\).
02

Substitute Values into the Formula

Substitute the given values of \(a\), \(b\), and \(c\) into the expression \(\sqrt{b^2 - 4ac}\). This results in: \(\sqrt{6^2 - 4 \times 1 \times 2}\).
03

Calculate \(b^2\)

Calculate \(b^2\) by squaring the value of \(b\): \(6^2 = 36\).
04

Calculate \(4ac\)

Compute the value of \(4ac\) by multiplying \(4\), \(a\), and \(c\): \(4 \times 1 \times 2 = 8\).
05

Subtract \(4ac\) from \(b^2\)

Subtract the result from Step 4 from the result in Step 3: \(36 - 8 = 28\).
06

Evaluate the Square Root

Find the square root of the result from Step 5: \(\sqrt{28}\).
07

Simplify the Square Root

Simplify \(\sqrt{28}\) by expressing it in terms of its factors. \(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}\).

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.

Discriminant
In quadratic expressions, the discriminant is a key component used to assess the nature of the roots of the equation. Specifically, for a quadratic equation in the standard form expression as \(ax^2 + bx + c = 0\), the discriminant is represented by \(b^2 - 4ac\).
  • If the discriminant is positive, this indicates two distinct real roots.
  • If the discriminant is zero, there is exactly one real root, also known as a repeated or double root.
  • Conversely, a negative discriminant means that there are no real roots, instead, the roots are complex numbers.
Understanding the discriminant helps one鈥攁t a glance鈥攑redict the kinds of solutions they are dealing with, whether it be real or complex numbers. So, before diving into solving a quadratic equation, evaluating this part can be quite informative.
Square Root Calculation
Calculating square roots is fundamental in simplifying quadratic expressions, especially under the discriminant.
  • To find the square root of a number \(n\), we look for a number \(m\) such that \(m^2 = n\).
  • Some numbers, like 4 or 9, are perfect squares, meaning they have integer square roots. For example, \(\sqrt{4} = 2\), because \(2^2 = 4\).
  • For numbers that are not perfect squares, like 28 in our exercise, the result will be an irrational number, which cannot be expressed as a simple fraction.
To find a satisfactory approximation or exact value of such square roots, it can be beneficial to express the number as a product of a perfect square and another number to simplify further.
Simplifying Radicals
Simplifying radicals involves rewriting a square root expression in its simplest form. This process often involves separating the radical into integer factors to streamline calculations and clarity. Consider \(\sqrt{28}\):
  • Factor 28 into its prime components or convenient products: 28 can be expressed as \(4 \times 7\).
  • Since 4 is a perfect square, it can be simplified as \(2^2\).
  • Break the square root across the multiplication: \(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7}\).
  • Solve for the square root of the perfect square \(\sqrt{4} = 2\), thus yielding \(2\sqrt{7}\).
Simplifying radicals helps in reducing the complexity of expressions and often makes further mathematical operations easier to manage.

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

Solve by completing the square. See Section 8.1. $$ x^{2}+4 x=12 $$

A common equation used in business is a demand equation. It expresses the relationship between the unit price of some commodity and the quantity demanded. For Exercises III and 112 , p represents the unit price and x represents the quantity demanded in thousands. A manufacturing company has found that the demand equation for a certain type of scissors is given by the equation \(p=-x^{2}+47 .\) Find the demand for the scissors if the price is \(\$ 11\) per pair.

If you are borrowing money at a rate of \(10 \%,\) would you prefer the interest to be simple or compound? Explain your answer.

Use the formula \(A=P(1+r)^{t}\) to solve Exercises 75 through 78 . See Example \(9 .\) Find the rate \(r\) at which \(\$ 800\) compounded annually grows to \(\$ 882\) in 2 years.

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=x^{2}+1 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.