/*! 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 69 Find all solutions of the equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 69

Find all solutions of the equation and express them in the form \(a+b i.\) $$\frac{1}{2} x^{2}-x+5=0$$

Short Answer

Expert verified
The solutions are \(1 + 3i\) and \(1 - 3i\).

Step by step solution

01

Identify the type of equation

The given equation is quadratic: \(\frac{1}{2}x^2 - x + 5 = 0\). We need to find its solutions.
02

Write the equation in standard form

To make it easier to solve, multiply the entire equation by 2 to eliminate the fraction: \(x^2 - 2x + 10 = 0\). This is now in standard quadratic form: \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = -2\), and \(c = 10\).
03

Calculate the discriminant

The discriminant \(\Delta\) is calculated using the formula \(b^2 - 4ac\). Substitute \(a\), \(b\), and \(c\): \((-2)^2 - 4 \times 1 \times 10 = 4 - 40 = -36\).
04

Determine solution type based on discriminant

Since the discriminant \(\Delta = -36\) is negative, the quadratic equation has two complex solutions.
05

Use the quadratic formula to find solutions

The quadratic formula is \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\). Substitute \(a = 1\), \(b = -2\), and \(\Delta = -36\) into the formula: \(x = \frac{-(-2) \pm \sqrt{-36}}{2 \times 1}\) \(= \frac{2 \pm \sqrt{-36}}{2}\).
06

Simplify the solutions

Since \(\sqrt{-36} = 6i\), substitute back into the formula: \(x = \frac{2 \pm 6i}{2}\). Simplify both solutions: \(\frac{2 + 6i}{2} = 1 + 3i\) and \(\frac{2 - 6i}{2} = 1 - 3i\).
07

Write the solutions in the form of \(a + bi\)

The solutions are \(1 + 3i\) and \(1 - 3i\), both in the required form \(a + bi\).

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 where the highest degree of the variable is 2. This means that the variable, often represented as \(x\), is squared. A standard quadratic equation takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Here鈥檚 a simple way to understand the parts of a quadratic equation:
  • \(a\): This is the coefficient of \(x^2\), crucial for determining the equation's shape.
  • \(b\): Known as the linear coefficient, it is associated with the \(x\) term.
  • \(c\): This is the constant term that shifts the graph up or down on the y-axis.
Quadratic equations often appear in various mathematical and real-life problems, such as calculating areas, determining projectile paths, and optimizing values. Solving them involves finding the values of \(x\) that make the equation equal to zero.
Discriminant
The discriminant is a critical component of the quadratic equation, used to determine the nature of its solutions. For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is given by the expression \(b^2 - 4ac\). The value of the discriminant gives insight into the type and number of solutions the quadratic equation has.
  • If \(\Delta > 0\), the equation has two distinct real solutions.
  • If \(\Delta = 0\), there is exactly one real solution, also referred to as a repeated root.
  • If \(\Delta < 0\), as in our case \(-36\), the equation results in two complex solutions, which are conjugates of each other.
Understanding the discriminant helps us anticipate whether the solutions will be real numbers or involve complex numbers and can indicate whether further simplification will be necessary.
Quadratic Formula
The quadratic formula is a universal method used to find solutions, also known as roots, of any quadratic equation. It is derived from completing the square and is expressed as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is incredibly handy, especially when factoring is difficult or impossible. Here is how it works:
  • "\(+/-\)" indicates there are typically two solutions: one for the addition and one for the subtraction.
  • "\(-b\)" begins the process by shifting the roots opposite of \(b\)'s influence.
  • The square root term \(\sqrt{b^2 - 4ac}\) incorporates the discriminant, controlling whether the solutions are real or complex.
  • "\(/2a\)" normalizes the result according to the quadratic term鈥檚 coefficient.
In the provided solution, since the discriminant \(-36\) is negative, the square root becomes complex, leading to the complex solutions \(1 + 3i\) and \(1 - 3i\). This illustrates the power of the quadratic formula to address not only real but also complex 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

Find all horizontal and vertical asymptotes (if any). $$r(x)=\frac{x^{3}+3 x^{2}}{x^{2}-4}$$

Show that the polynomial does not have any rational zeros. $$P(x)=2 x^{4}-x^{3}+x+2$$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=x^{4}-2 x^{3}-9 x^{2}+2 x+8 ; \quad a=-3, b=5$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6$$

Use transformations of the graph of \(y=\frac{1}{x}\) to graph the rational function, as in Example 2. $$r(x)=\frac{1}{x+4}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.