# 0.5 Quadratic equations

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This module is part of a collection of modules that were developed to support laboratory activities in a Precalculus for PreEngineers (MATH 1508) at the University of Texas at El Paso. Contained in this module of applications of quadratic equations in various fields of engineering and science. These include the motion of an object under constant acceleration, quantitative management, and break-even analysis.

## Introduction

Quadratic equations play an important role in the modeling of many physical situations. Finding the roots of quadratic equations is a necessary skill. Being able to interpret these roots is an important ability that is important in understanding physical problems. In this module, we will present a number of applications of quadratic equations in several fields of engineering.

## Determining the roots of quadratic equations

A quadratic equation has the following form

${\text{ax}}^{2}+\text{bx}+c=0$

Because a quadratic equation involves a polynomial of order 2, it will have two roots. In general, a quadratic equation will either have two roots that are both real or have two roots that are both complex. For the present module, we will restrict our attention to quadratic equations that have two real roots.

There are three methods that are effective in solving for the roots of a quadratic equation. They are:

• Solution by factoring
• Solution by completing the square
• Solution by the quadratic formula

The applications that follow will include examples of each of these three methods of solution.

## Motion of an object under uniform acceleration

We will begin our study of quadratic equations by considering an application that you will likely encounter later in physics and mechanical engineering classes. Let us consider an object that is subject to a uniform acceleration. By uniform, we mean an acceleration that is constant. Such an object might be an automobile, an aircraft, a rocket, etc. The motion of an object subjected to uniform acceleration can be expressed mathematically by the following equation.

$s\left(t\right)=\frac{1}{2}a{t}^{2}+{v}_{0}t+{s}_{0}$

where s ( t ) represents the position of the object as function of time t ,

a represents the constant acceleration of the object,

v 0 represents the value of the object’s velocity at time t = 0, and

s 0 represents the position of the object at time t = 0.

An equation of this sort is called an equation of motion . We will illustrate its use in the following exercise.

Example 1: For our first example, let us consider a dragster on a drag strip of length one-quarter mile. For time t<0, the dragster is at rest at the starting line. At time = 0, the driver depresses his gas pedal to produce a uniform acceleration of 50 m/s 2 . Under these conditions, how far will the dragster travel in 1 second?

Because the dragster travels in a horizontal direction, we will represent its distance from the starting point as a fuction of time as x ( t ). We also know that the value for the acceleration ( a ) is 30 m / s 2 . We can incorporate these changes in equation (1) to produce a new equation of motion for the dragster.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution   By Lakeima Roberts      By 