/*! 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 81 Use the discriminant to determin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 81

Use the discriminant to determine the number of real soIutions of the equation. Do not solve the equation. $$x^{2}+2.20 x+1.21=0$$

Short Answer

Expert verified
The equation has exactly one real solution.

Step by step solution

01

Identify quadratic coefficients

The given quadratic equation is \(x^2 + 2.20x + 1.21 = 0\). Here, we need to identify the coefficients: \(a = 1\), \(b = 2.20\), and \(c = 1.21\).
02

Write the formula for the discriminant

The formula for the discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is \(D = b^2 - 4ac\).
03

Substitute the coefficients into the discriminant formula

Substitute \(a = 1\), \(b = 2.20\), and \(c = 1.21\) into the discriminant formula: \[D = (2.20)^2 - 4 \times 1 \times 1.21\].
04

Calculate the discriminant

Calculate the values: \((2.20)^2 = 4.84\) and \(4 \times 1 \times 1.21 = 4.84\). So the discriminant \(D = 4.84 - 4.84 = 0\).
05

Determine the number of real solutions

Since the discriminant \(D = 0\), it indicates that there is exactly one real solution for the quadratic equation. This is because a discriminant of zero implies the presence of one real, repeated solution (or a double root).

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 Equation
A quadratic equation is a type of polynomial equation that involves a variable raised to the second power. It usually takes the form \( ax^2 + bx + c = 0 \), where:
  • \( a \), \( b \), and \( c \) are constants known as coefficients
  • \( x \) represents the variable or unknown
  • \( a eq 0 \) because if \( a = 0 \), the equation becomes linear, not quadratic
The goal when dealing with quadratic equations is often to find the values of \( x \) that make the equation true. These values are called the "solutions" or "roots" of the equation.

Quadratic equations can have different types of solutions. These solutions depend on the value of the discriminant, which is a particular value derived from the coefficients \( a \), \( b \), and \( c \).
Real Solutions
When we talk about the "real solutions" in a quadratic equation, we refer to the values of \( x \) that are real numbers, as opposed to imaginary or complex numbers. To determine the number of real solutions, we employ the discriminant \( D \), found using the formula:
  • \( D = b^2 - 4ac \)
Here鈥檚 how the discriminant helps:
  • If \( D > 0 \), there are two distinct real solutions.
  • If \( D = 0 \), there is exactly one real solution, known as a repeated or double root.
  • If \( D < 0 \), there are no real solutions; instead, there are two complex solutions.
For example, in the equation from the exercise, we calculated the discriminant \( D = 0 \). This tells us that the equation has exactly one real solution.

Understanding the number and type of solutions can provide valuable insights into the nature of the quadratic equation.
Coefficients
Coefficients play a crucial role in the structure and solution of quadratic equations. They are the numbers \( a \), \( b \), and \( c \) in the standard form of the equation \( ax^2 + bx + c = 0 \). Each coefficient has a specific role:
  • The coefficient \( a \) is associated with \( x^2 \) and determines the parabola's "width" and "direction" (upward if \( a > 0 \), downward if \( a < 0 \)).
  • The coefficient \( b \) is linked with \( x \) and influences the parabola's "position" on the horizontal axis.
  • The constant \( c \) represents the parabola's "y-intercept," the point at which it crosses the y-axis.
In the exercise, identifying the coefficients (\( a = 1 \), \( b = 2.20 \), and \( c = 1.21 \)) was the first step in using the discriminant to find the number of real solutions.

Proper identification and understanding of these coefficients are key to successfully analyzing and solving quadratic equations.

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

Perform the indicated operations and simplify. $$(2 x-5)\left(x^{2}-x+1\right)$$

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(2 y+5)(2 y-5)$$

Factor the expression completely. $$x^{6}+64$$

Factor the expression by grouping terms. $$x^{3}+x^{2}+x+1$$

\(x^{4}+a x^{2}+b\) A trinomial of the form \(x^{4}+a x^{2}+b\) can sometimes be factored easily. For example, $$ x^{4}+3 x^{2}-4=\left(x^{2}+4\right)\left(x^{2}-1\right) $$ But \(x^{4}+3 x^{2}+4\) cannot be factored in this way. Instead, we can use the following method. \(x^{4}+3 x^{2}+4=\left(x^{4}+4 x^{2}+4\right)-x^{2} \quad \begin{array}{l}\text { Add and } \\ \text { subtract } x^{2}\end{array}\) Factor perfect \(=\left(x^{2}+2\right)^{2}-x^{2}\) \(=\left[\left(x^{2}+2\right)-x\right]\left[\left(x^{2}+2\right)+x\right] \quad \begin{array}{l}\text { Difference of } \\ \text { squares }\end{array}\) \(=\left(x^{2}-x+2\right)\left(x^{2}+x+2\right)\) Factor the following, using whichever method is appropriate. (a) \(x^{4}+x^{2}-2\) (b) \(x^{4}+2 x^{2}+9\) (c) \(x^{4}+4 x^{2}+16\) (d) \(x^{4}+2 x^{2}+1\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Pure 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.