/*! 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 23 Solve each of the following quad... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 23

Solve each of the following quadratic equations, and check your solutions. $$y^{2}-2 y=-19$$

Short Answer

Expert verified
The solutions are complex: \( y = 1 \pm 3\sqrt{2}i \).

Step by step solution

01

Write the Equation in Standard Form

Given the quadratic equation is \( y^2 - 2y = -19 \). To write it in standard form, move all terms to one side of the equation. Add 19 to both sides: \( y^2 - 2y + 19 = 0 \).
02

Identify the Coefficients

With the equation in standard form \( y^2 - 2y + 19 = 0 \), identify the coefficients: \( a = 1 \), \( b = -2 \), and \( c = 19 \).
03

Apply the Quadratic Formula

The quadratic formula is \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the coefficients: \( a = 1 \), \( b = -2 \), \( c = 19 \).
04

Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \): \( (-2)^2 - 4 \times 1 \times 19 = 4 - 76 = -72 \).
05

Evaluate the Square Root of the Discriminant

Since the discriminant is negative, \( \sqrt{-72} = \sqrt{72}i = 6\sqrt{2}i \) where \( i \) is the imaginary unit.
06

Compute the Solutions Using the Quadratic Formula

Substitute the values back into the quadratic formula: \( y = \frac{-(-2) \pm 6\sqrt{2}i}{2 \times 1} \).Thus, the solutions are \( y = \frac{2 \pm 6\sqrt{2}i}{2} \), which simplifies to \( y = 1 \pm 3\sqrt{2}i \).
07

Interpret the Solutions

The solutions \( y = 1 + 3\sqrt{2}i \) and \( y = 1 - 3\sqrt{2}i \) are complex numbers, indicating there are no real number solutions for the given 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.

Imaginary Numbers
When solving quadratic equations, especially with a negative discriminant, imaginary numbers become central. An imaginary number is formed when we take the square root of a negative number. This happens because the product of two identical real numbers is always positive. Thus, we introduce 'i' as the imaginary unit, representing the square root of -1, where \( i^2 = -1 \).
  • For instance, \( \sqrt{-1} = i \).
  • The complex number system extends the real number system and includes numbers such as 3 + 4i or -2i.
In the context of the given equation, after finding a negative discriminant, we encountered \( \sqrt{-72} \), which means we had to use 'i' to express it as \( 6\sqrt{2}i \). Imaginary numbers thus help solve equations that would otherwise lack solutions in the real number system.
Quadratic Formula
The quadratic formula is a fundamental tool in mathematics for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), allows us to find the roots (solutions) of the equation by simply plugging in the coefficients a, b, and c.
  • If the discriminant (\(b^2 - 4ac\)) is positive, you'll get two real solutions.
  • If it's zero, one real solution exists.
  • If it's negative, as in our example, you'll find two complex solutions.
Remember, the quadratic formula is a reliable method because it works for all types of quadratic equations regardless of the discriminant's sign.
Discriminant
The discriminant, found within the quadratic formula, is crucial for predicting the nature of solutions we will obtain. Calculated as \( b^2 - 4ac \), it tells us whether our equation has real or complex solutions.
  • A positive discriminant means two distinct real solutions.
  • Zero means exactly one real (repeated) solution.
  • A negative discriminant, as in this exercise, yields two complex solutions.
In our example, the discriminant was \(-72\), indicating complex solutions. This negative value suggested non-real outputs, leading us to look at imaginary numbers for our final answers.
Complex Solutions
Complex solutions in quadratic equations arise when the discriminant is less than zero. These non-real solutions combine a real part and an imaginary part. A complex number takes the form \( a + bi \), where 'a' is the real part, and 'bi' is the imaginary part.
When solving the equation from the example, we obtained the solutions \( y = 1 \pm 3\sqrt{2}i \). Here, 1 is the real part, and \( 3\sqrt{2}i \) comprises the imaginary portion.
  • Complex solutions always come in conjugate pairs: \( a + bi \) and \( a - bi \).
  • These solutions mean there is no point where the equation crosses the x-axis, useful in graphing functions.
Understanding complex solutions is vital in advanced mathematics, physics, and engineering, where they describe phenomena beyond real number scenarios.

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 hotel charges $$\$ 98.50$$ for a room with one guest. If there is more than one guest, then the charge for the room is expressed as the function \(f(n)=98.50+20(n-1)\), where \(n\) is the number of guests. What are the charges when there are 2 guests, 3 guests, 1 guest, and 4 guests?

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ (7 i)(8 i) $$

A boat rental agency charges $$\$ 50$$ plus $$\$ 35$$ per hour to rent a fishing boat. Therefore, the charge for renting a boat is a function of the number of hours \(h\) the boat is rented and can be expressed as \(f(h)=50+35 h\). Compute \(f(5), f(10), f(1)\), and \(f(3)\).

Solve each of the following quadratic equations, and check your solutions. $$y^{2}+8 y=-24$$

For Problems \(13-34\), add or subtract the complex numbers as indicated. $$ \left(\frac{1}{6}+\frac{5}{4} i\right)-\left(-\frac{1}{3}-\frac{3}{5} i\right) $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.