Angzzas Sari MOHD KASSIM, Chee-Ming CHAN, Suraya Hani ADNAN, Hong-Yin LAM, Abdul Fathul Hakim ZULKIFLI, Muhammad Farid SHAARI and Ishak BABA
Faculty of Engineering Technology, Universiti Tun Hussein Onn Malaysia (MALAYSIA) [email protected]
Abstract
Committed to the highest standard of quality in the provision of higher education in the professional field of civil engineering technology, a comprehensive Quality Management System (QMS) was developed as the essential framework to effectively oversee the QA&QC processes as well as to meet the stakeholders’ expectations at all levels. The fully online integrated system captures real-time data for continual monitoring anytime, anywhere. It comprises of 3 primary sub-systems, namely the Student Assessment System (SAS), OBE System (OBESys) and the Student Information System (SMP), which are harmonized with the greater university-wide anchor system named Total Campus Integrated System (TCIS), incorporating all ancillary arms in the operation of the institution.
This paper explains on the operational environment supported by the various centralized offices related to academic and non-academic development of students, such as the Academic Management Office, Information Technology Centre, University and Industry Relation Office, among others. The policies and mechanisms adopted for attracting and retaining outstanding staff on board is also discussed, in conjunction with a constructive and supportive leadership at the helm. In addition, assessment mechanisms for both staff and students are elaborated to show the holistic monitoring and close-the-loop improvement exercises in place for effective QA&QC. Besides, participation of the industry via programme review, industrial engagement at course level, research and consultation projects and others are highlighted to ensure relevance of the engineering technology programme with current market needs and expectations. In a nutshell the QMS executed is shown to not only coordinate but consolidate the various work processes in the machinery of the University, ensuring the seemingly stand-alone offices to have their outputs seamlessly advanced to the overall good performance of the academic programme.
Introduction
Quality assurance and control as well as quality management systems are not modern notions. Ancient Egyptians defined quality as ‘a sign of perfection’, such as demonstrated in the architectural precision and aesthetics of the royal tombs of pyramids (Elshennawy, 2004). In the academia, quality could be interpreted as relating to academic standards (Cheng, 2011), though Brink further elaborated that quality could be the answer to the question of ‘is it good?’, while standards address the question of ‘is it good enough?’ (Brink, 2010). However, quality in education is closely related to 4 main purposes: preparation for sustainable employment, preparation for life in a democratic society, personal development as well as development and maintenance of a broad knowledge base, i.e. lifelong learning (Bergen, 2011). Assessment of stakeholders’ satisfaction on the educational processes, organisation and outcomes would serve well for institutions of higher learning to take necessary and timely preventive or corrective measures to upkeep the quality of the programmes offered, though the reliability of the inputs depend very much on the techniques and instruments adopted to acquire data (Ham & Hayduk, 2003).
Continual Quality Improvement
For continual improvement, the educational programme is evaluated regularly to keep in line with current development and to align it with the needs of stakeholders. In this regards, the Faculty, as well as the University have applied a comprehensive PDCA (Planning, Doing, Checking and Acting) cycle to ensure successful execution of all processes related to PEOs, PLOs and CLOs. The PDCA cycle of the overall quality management at different levels. At the University level, the PDCA cycle is fundamentally framed on the basis of the University’s long term strategic plan (UTHM Blueprint). The University’s blueprint as outlined in the central policy framework is supported by implementation of relevant processes involving the Faculty, its departments, units and individual staff members. The blueprint is revised accordingly whenever deemed necessary in order to ensure the set targets are systematically achieved. The quality assurance at this level is ascertained with annual internal auditing processes, i.e., the internal auditing, which are managed by centre for Academic Development and Training (CAD), and external auditing, i.e. ISO by SIRIM. The auditing ensures that the structure and quality of task implementation complies with set procedures and regulations. In addition, the University has also developed the Internal Complaints and Suggestions Channel System (SACAD), as one of the channels for students and staff to give feedbacks concerning the University at any time.
From Programme Outcomes To Quality Improvement
Figure 1 summarizes the overall inter-relationship between PEOs, PLOs and CLOs, which collectively contribute to the assessment of PLO attainment of the Programme. Represented in a pyramidal arrangement from bottom-up are the tasks / activities carried out and assessed in each course of the Programme, followed by the CLOs for each of the course, which cumulatively build up the respective PLOs or POs at a higher order, and finally culminating in the attainment of PEOs measurable after a certain lapse from graduation. On the left wing of the pyramid are listed the instruments of assessment for each component, namely SAS for gauging the attainment of CLOs at course level, OBESys for measuring the achievement level of PLOs cumulatively from the contributing CLOs throughout a student’s study, and finally Tracer Study for evaluating the graduates’ attainment of PEOs 3-5 years after completion of the Programme. Assessment data collected with the aid of the instruments are then analysed to identify the strength and weaknesses of the components in formulating short and long term improvement strategy for the Programme as a whole, i.e. close-loop CQI practices. Understandably, the intricately related system requires the change in one component to have an influence on the others to various degrees.
It follows that a discourse on the attainment of PLOs alone cannot illustrate the full picture of the effective further development and improvement of the Programme. The overall structure of the Programme would be affected in one way or another for every change made to any of the component. To address the particular concern of Criterion 2 in terms of adoption of the POs’ assessment outcomes for improvement of the Programme, it is apparent from Figure 1 that the attainment level for each PLO for individual students can be retrieved and reviewed via the OBESys. Indeed, performance of the entire cohort of students can also be analysed for comparison purposes simply by the sorting function of the OBESys. These findings are then used for identifying unforeseen shortcomings, untapped potential as well as unchartered waters for betterment of the Programme. Also, these reports serve as useful references for reviewers of the Programme (e.g. External Examiner) to gain valuable insights of the Programme, guiding them to make informed decisions or recommendations in their reviews.
All these help make the process of making relevant and spot-on improvement to the Programme a timely and orderly one. The real-time data management of OBESys allows students’ performance to be monitored closely, particularly in terms of the time-dependent, cumulative attainment of PLOs. In other words, review of the PLOs’
attainment need not be postponed till the end of the student’s study, but can be accessed and analysed at any point of time with a display of the accrued merits for each PLO. The following discourse details out the infrastructure for quality assurance and control of academic programmes in the University, primarily under the umbrella of quality management system, i.e. MS ISO 9001: 2008. In general, UTHM implements Quality Academic Practices under the systematic approach outlined in MS ISO 9001: 2008. Through the implementation of MS ISO, review of the academic curriculum in terms of stakeholders’ input is conducted in accordance to RPK-02 Curriculum Review Procedures. The Tracer Study is also conducted to serve as an informative tool to ascertain the employability of graduates, preferred or dominant fields of work, potential career prospect, among others for continuous improvement of the Programme.
Figure 1. Hierarchical organisation and cross-relationship between the components for CQI at each level.
Continual Quality Improvement (CQI) is necessary and vital to ensure the sustained development and betterment of an academic programme with regular self-checks. It is necessarily implemented at all levels of a quality management system, such as found in the academia in providing value-for-money academic programmes of
assured quality. The improvement strategy and approaches must encompass the entire ecosystem of the institution, especially in the Programme’s fundamental architecture, founding philosophy and aspirations. Therefore, the CQI processes cut across the board from the Programme to the Course levels as described below.
At the Programme level, CQI is implemented via various tools including Faculty Academic Committee meetings, OBE performance monitoring, Internal Quality Audit reports, University Strategic Action Plan, stakeholders’
review meetings and external examiner reviews. At Course level, CQI is required for cases where students score less than 50% of the total marks. The CQI activities are subjected to the creativity and innovation of respective academic staff with considerations of the suitability. These activities could be in the forms of extra classes with additional exercises and/or teaching materials, individual tutoring sessions, as well as revisions and discussions on completed assignments and tests.
All in all the attainment of PLOs are shown to be supported by the assessment and attainment of CLOs from the downstream end, and supporting the evaluation of PEOs attainment on the upstream end (refer Figure 1). The CQI cycle at Programme and Course levels effectively extracts insightful data and findings as feeder information for review, revision and overall improvement of the Programme as a whole.
Evidence-Based Attainment Of Programme Outcomes
Meeting the PLOs via collective attainment of CLOs in the respective Courses are substantiated by a number of documented evidence. These evidences can be broadly divided into 2 categories: materials by academic staff and students (Figure 2). As shown in Figure 2, the materials prepared by the academic staff for a Course include the question papers and answer schemes prepared for tests and final examinations, with detailed cognitive domain analysis on differentiation of the difficulty levels as per Bloom’s taxonomy. It also encompasses structured questions or problems for course assignments and projects (PBL exercises), questions and answers for oral examination of laboratory work, suitably themed Bachelor Degree Project (BDP) titles and scope, and other tasks assigned to students throughout the duration of study. From the students’ end, evidence collected include project reports or assignments with accompanying visual aids in various formats (e.g. brochure, poster, video, model, etc.), as well as compiled in threes of the best, moderate and low achievers: answer scripts for tests and final examinations, laboratory reports, assignments, and other written work. Other evidence in this category can also be found in the reports and logbooks for Bachelor Degree Project, Industrial Training, field work, plant and site visits as well as industrial talks or seminars.
Taking into account that a holistic graduate should be equipped with both academic and non-academic skills or competencies, evidence of students’ effort contributing to the attainment of PLOs also include records of the various activities organised by and/or participated in the Faculty and Departmental Student Clubs, university-wide student associations, sports clubs, uniform bodies and others. Students are encouraged to take part in national and international levels competitions too, such as language, cultural and sports events, both co-organised by the University or entirely managed by outside entities. These go on to show the academic prowess and extra-curricular aptitude of the students, leading to the attainment of PLOs which are essentially an epitome of the desired product of the Programme, i.e. an all-rounder graduates with a balanced flair to function as a member of profession and society in general.
Figure 2. Evidence substantiating the attainment of programme outcomes.
Cqi For Quality Assurance And Control
The commitment of the University to quality assurance is demonstrated by the Quality Policy statement that strives to provide quality academic programmes that meet the stakeholders’ satisfaction through excellent organizational
maintain effective quality management satisfying MS ISO 9001:2008 certification standards.
The University Quality Objectives are implemented to ensure the achievement of the University’s vision and mission. During Management Review Meeting, the Faculty Quality Objectives were also consistently reviewed.
UTHM has been awarded the ISO 9001:2008 - Management of Academic Programmes. In 2004, 2008 and 2010, UTHM has passed the surveillance Audit excellently and the award was extended to 2016. There are now 68 procedures in total which are continually amended to accommodate new changes. Since practiced in 2001, the ISO 9001:2008 has become the motivation of quality academic programmes management.
For Continual Quality Improvement (CQI), annual ISO audit and academic audit are conducted for all Faculties’
academic programmes. Three processes related to academic programme were proposed to be reviewed in the recent internal audit to ensure quality programme development and to address the requirements of OBE and stakeholders, namely academic programme development, academic programme review and implementation of lecture processes.
a. System for Examination Regulation
UTHM has established procedures for preparation of the final examination papers, in accordance with the ISO standard. The Academic Management Office (PPA) is responsible to monitor the workflow of the final examination regulation, beginning with the preparation of examination paper (RPP-12 Procedure for Final Examination Paper Preparation), assessment of the examination paper (RPP-13 Procedure for Final Examination Paper Specifications), preparation of the final examination (RPP-14 Procedure for Final Examination Service), and the submission of the final examination results (RPP-15 Procedure for Final Examination Collecting, Inserting, Checking of Marks). At the Faculty level, the Faculty Academic Office is responsible in monitoring academic staff’s adherence to the work processes.
b. System of Assessment for Examinations, Projects and Industrial Training i. System of Assessment for Examinations
At the Faculty level, the Examination Committee is responsible to manage the preparation, assessment and moderation of the examination papers. The questions for final examinations are Prepared by the academic staff according to level of Bloom’s taxonomy that has been set according to the CLOs of the particular course. Table of Specification is used to ensure that the questions are correctly mapped according to the CLOs, during formative and summative assessments so that every CLOs and consecutively, PLOs could be assessed i.e. constructively aligned. In ensuring the quality of final examination questions, the drafts of the final examination paper undergo two levels of moderation processes. At first, the draft is vetted at the departmental level using the Vetting Form of Test and Final Examination, and should fulfil the following requirements:
• Compliance to ISO (RPP-15)
• Proper usage of terms
• Suitability of questions mapped to CLOs and learning taxonomies
• Suitability of examination duration with the course’s number of credits
• Suitability of Level of Bloom’s taxonomy ii. System of Assessment for Final Year Projects
The procedures consist of several aspects including submission of Bachelor Degree Project title and proposal by students, appointment of supervisor, students briefing on Bachelor Degree Project, and approval of project titles by Head of Panels, supervision, and the Bachelor Degree Project assessment. Since Semester 1, Session 2016/2017, ePSM system, an online Bachelor Degree Project management system, was developed by Centre of Information Technology (PTM) for facilitating the implementation of Bachelor Degree Project. The submission of Bachelor Degree Project title and proposal by the students is conducted using the ePSM. In addition, the marks from supervisor and examiner panels are recorded in the system, before the coordinator of Bachelor Degree Project transfers the marks to SAS upon marks moderation.
ii. Systemof Assessment for Industrial Training
The assessment of students’ performances for industrial training are divided into two stages, namely, the industry on-site visits by the Faculty supervisor and upon the completion of industrial training, with reference to RPP-11 Procedure of Industrial training (Supervision & Evaluation) and the University’s Industrial Training Guideline.
c. System for Teaching and Learning
In order to ensure the quality of the teaching and learning process, each academic staff is provided with the Desk File, containing ISO procedures as a clear guideline and reference with their respective scope of work. Apart from appointing relevant stakeholders, the Faculty is continuously improving the quality of each programme by benchmarking exercises so the standard is appropriate or even better than programmes offered by other renowned
local and international universities. To date three benchmarking exercises were conducted with universities in Australia and also locally at UMP and UNIKL. Findings from the benchmarking exercises proved that the Programme is of appropriate standard and quality, locally and internationally. Good practices from other universities, including environmental engineering technology laboratory practices such as pilot scale system for water, wastewater treatment system as well as air pollution control devices were also introduced in the teaching and learning to enhance student’s skills and competencies.
Conclusions
Quality assurance and quality control are essential to regular monitoring of an academic programme, as well as serving to promote continual improvement and upgrading of the supporting system. Inputs from various offices are consolidated to gauge, rectify and enhance the academic processes involved in the smooth running of the Programme. Such an integrated quality management system helps eliminate hiccups due to human errors, delays and unforeseen logistical glitches, while keeping real-time records of every element in the Programme even after the student has left the University.
Acknowledgements
Kudos to the Faculty’s AA Team (academic and supporting staff) for the successful ETAC accreditation of programmes late last year.
References
Elshennawy, A (2004). Quality in the new age and the body of knowledge for quality engineers. Total Quality Management & Business Excellence International, vol. 15, no. 5, pp. 603-614.
Cheng, M. (2011). Transforming the learner versus passing the exam: Understanding the gap between academic and student definitions of quality. Quality in Higher Education, vol. 17, no. 1, pp. 3-17.
Brink, C. (2010).Quality and standards: Clarity, comparability and responsibility Quality in Higher Education, vol. 16, no. 2, pp. 139-152.
Bergen, S. (2011)“Building trust,” Quality and Trust: At the Heart of What We Do. European Quality Assurance Forum, pp. 52-57.
Ham, L. & Hayduk, S. (2003). Gaining competitive advantages in higher education: Analysing the gap between expectations and perceptions of service quality, International Journal of Value-based Management, vol.
16, no. 3, pp. 223-242.
A Different Approach To Reasoning In Mathematics And Accounting Course: Accounting Department Sample∗
Hüseyin Soydaş
Kocaeli University Kocaeli Vocational School Turkey [email protected]
Abstract
Original reasoning rational remains by means of aim toward reason. Triggering the expressive control of enchanted then unknown (cutting-edge additional arguments original reasoning) remains unique of the newest then influential tendencies cutting-edge occupational university teaching. We stretch now happening an examination of the organizations aimed at original reasoning have enough money toward occupational university scholars. This education usages the perfect industrialized through Lithner (2008) which distinguishes amid derivative reasoning which remains connected toward repetition knowledge of procedures then ideas, then original reasoning which includes reasonable mathematically originated influences. The examination includes the inspection of minutes, projects then inspections rummage-sale cutting-edge rudimentary mathematics then accounting course cutting-edge Accounting Department of Kocaeli College Kocaeli occupational university by the opinion toward categorizing the kinds of reasoning predictable of the scholars. By way of healthy by way of explanation our usage of lithner’s technique, we deliberate his appropriateness by way of a instrument aimed at categorizing reasoning chances cutting-edge communal discipline department’s mathematics course. Development in educational works of participants apply of rational suggestions supports the study’s style work on the data
Keywords: vocational school, mathematical reasoning, accounting department Introduction
School applications of participants math showing a positive effected theoretical background of math demonstrating, detailing registering skill in math, for more powerful presentation of the various features of participants skills in math demonstrating.
The supplementary way involved participants estimation of math evidences and rational learning of various proof methods (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008; Alcock & Weber, 2005; Antonini & Mariotti, 2008; Harel & Sowder, 1998; Selden & Selden, 2003; Stylianides & Stylianides, 2009). The abovementioned writing restricted ections of reason exercise, moreover, didn’t discover the use of numerical examples as an alternate of reasonable exercise not the power of this teaching on participants justification of math evidences and rational information of various evidence methods (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008; Bergqvist, E. 2007.; Aydin At All, 2018; Şeneldir At All, 2017; Deer, 1969;
Durand-Guerrier, 2003; Epp, 2003; Mueller, 1975; J. L. Platt, 1967). Lithner (2008) describes reasoning by way of ‘the streak of supposed accepted toward crop declarations then spread deductions cutting-edge job (problematic)-solving’. This meaning consumes together tall then low-quality influences then remains not lone limited toward official evidences of propositions. The outline remains valuable cutting-edge learning the rational procedures wanted toward resolve glitches cutting-edge rudimentary mathematics then accounting course, anywhere frequently evidences remain not assumed nonetheless scholars remain predictable toward brand reasonable influences then deductions. Lithner distinguishes amid derivative reasoning which remains connected toward repetition knowledge then imitation of procedures then original reasoning which includes reasonable mathematically-founded influences. Cutting-edge this education, he remains practical this outline toward categorize the reasoning chances obtainable cutting-edge a variety of commercial mathematics obtainable cutting-edge Niğde Ömer Halisdemir communal discipline occupational university. We remain seeing together course aimed at professional then non-specialist scholars, by way of healthy by way of required then elective units (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K.
2008; Battal At All., 2017; Bostan, And Durmuş, 2016; Bostan, And Durmuş, 2017; Durmuş, 2016).
Rendering the teaching investigators, change after tall university (before school) toward college remains extensively recognized by way of a problematic procedure then scholars typically discovery that the change cutting-edge mathematics then accounting talks (physic then interaction) remains particularly problematic then difficult (Pointon, A., and C. Sangwin. 2003; Kılıçaslan 2014; Boesen Then Lithner, 2010). Scholars’ problems cutting-edge primary day originate after theoretical approach somewhat than real performance of mathematics educator (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008; Aydin At All, 2017a;
Aydin And Mutlu, 2013; Fucawa-Connelly, 2005). Scholars’ problems remain particularly by the ideas such by way of multifaceted statistics, singular purpose, derivate, copied, then intensely examining of meaning of these ideas then their extended then nonconcrete evidences. These rudimentary ideas of college mathematics remain too extensively rummage-sale after the scholars counting the departments such by way of manufacturing, physic, ecology, approximately communal agendas.
He remains a famous then putative mathematical quarrel that the education of mathematics remain influential stage through stage by the growth of nonconcrete rational services, then the unique of the chief drives of mathematics teaching remains toward impart mathematical reasoning (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008; Ball, D. L., Thames, M. H., Phelps, M. 2008.; Koparan At All, 2018; Clark And Lovric, 2009).
Conceptual framework
Mathematical evidences remain crops of mathematical proving. Though, we usage the period “mathematical showing” cutting-edge this education needed rummage-sale by resistant manufactures (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008;
Pointon, A., and C. Sangwin. 2003; Harel & Sowder, 1998; Weber, 2001).in order to categorize the proof-related skills that we describe in the future units, evidence information. The other skills had been planned as vital for participants evidence and verifying (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008; Görentaş And Yıldız, 1999; Bilgin At All, 2010; Yıldız And Görentaş; Bilgin And Görentaş, 2008; Alcock & Weber, 2005; Selden & Selden, 2003).
∗ A brief version of this article presented at INTE 2018
Cutting-edge this education approximately kind of queries will be scholars’ effort physical including homework problems.
Lithner (2008) clarified changes amid derivative then original reasoning. Derivative reasoning consumes binary chief kinds:
memorised then algorithmic. Cutting-edge instruction toward remain secret by way of memorised a reasoning process must consume complicated the next ladders (Lithner (2008): a) the plan excellent remains examined happening concentrating a whole response. B) the plan application wants toward to consume lone the registration.
The original reasoning organization container remain alienated hooked on binary subgroups (Dreyfus, 1991): native creative reasoning; then worldwide original reasoning. A problem set remains accepted “need native original reasoning” doubt he remains soluble by means of any procedure nonetheless the it wants toward be modified by the procedure nearby. A problem set remains accepted “need worldwide original reasoning” doubt he fixes not consume a answer that remains founded happening any procedure then needs original reasoning (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008; Of At All, 2018; Kahraman At All, 2018; Of At All, 2017; Tola At All, 2017). This kind of reasoning remains understood greatest frequently on the vocational school equal once scholars remain requested toward remember a meaning, toward national an axiom then a proof of exact proposition. Lithner (2008) notes a reasoning application imaginative when that has the next main types:
i. Novelty. A new reasoning sequence is applied
ii. Plausibility. Applying of opinions is necessary to support strategy implementation. It motivates students to think that the inferences are true or not
iii. Mathematical basis. Background level of student’s determinate the accomplishment of the strategies Method
Cutting-edge this education, we rummage-sale errands after mathematic course of communal discipline departments of the occupational university. The talks comprise four commercial mathematics units. These four units remain shaped after the mathematics and accounting course obtainable toward scholars. The information cutting-edge this education contains of the next kinds: talk minutes, schoolbooks, projects, inspection queries. We calm altogether the info by the collaboration of the unit speakers. The information examination of apiece unit remains now existence led through the writer of this object (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008).
This research method confirms dependability of the verification of the lecture elements from the various modules. We first made the examination through categorizing movements after model, cutting-edge instruction toward gain approximately knowledge then toward deliberate then decide happening our organization approaches. These events remain cutting-edge streak by persons obtainable through Lithner (2008). Lithner primary concept an answer toward the job then formerly likened toward the lecture minutes then model instances. By means of Lithner’s outline, we choose whether the job might remain resolved by means of algorithmic reasoning before whether original reasoning remains wanted. We originate that the greatest problematic choices worried the organization of errands hooked on the local creative reasoning or global creative reasoning groups, then consequently we set the outline as the method that: we first define that we would categorize a problem by way of local creative reasoning doubt the local creative reasoning remained founded happening an procedure, nonetheless scholars needed toward confirm the problem in subgroups (Bhaird, C., Nolan, B , O’shea, A, & , Pfeiffer, K. 2008).
Analyses and findings
We current approximately instances of errands secret by means of the Lithner reasoning outline. We distillate happening unique theme cutting-edge instruction toward remain intelligible then toward remain healthier talented toward liken groups. We reflect the theme of lucid reckonings, which remains significant cutting-edge numerous rudimentary mathematics course. Cutting-edge the sequence cutting-Cutting-edge query, the talk minutes then the model deliberate answers of lucid reckonings by means of the quadratic formulation by way of healthy by way of factoring then stretch instances which exemplify together approaches.
Question 1: solve the following rational equations, give your answers with various format if it is possible.
(a) 𝑥𝑥2 − 9𝑥𝑥 + 20 = 0;
(b) 𝑥𝑥2 − 2𝑥𝑥 - 2 = 0;
(c) 2𝑥𝑥 2 − 50 = 0;
(d) 𝑥𝑥2 − 18𝑥𝑥 + 81 = 0;
(e) 𝑥𝑥2 + 2𝑥𝑥 + 2 = 0;
(f) 𝑥𝑥2 − 10𝑥𝑥 + 170 = 14𝑥𝑥 + 30.
Analysis of the parts of the question:
In this question, students are expected to use the quadratic formula or factorization of the rational equations.
A) 𝑥𝑥2 − 9𝑥𝑥 + 20 = (𝑥𝑥 − 4)(𝑥𝑥 − 5), than the solutions are 𝑥𝑥 = 4,5;
B) by using of the quadratic formula of rational equations, than the final step of the solution is 𝑥𝑥 = 1±√3 , with other expression, 𝑥𝑥 = 2.732, and x = -0.732;
C) 2𝑥𝑥2 − 50 = 2(𝑥𝑥2 − 25) = 2(x – 5)(𝑥𝑥 + 5), so the solutions are 𝑥𝑥 = −5, 5;
D) 𝑥𝑥2 − 18𝑥𝑥 + 81 = (𝑥𝑥 − 9)2 , so there is just unique solution at 𝑥𝑥 = 9.
E) using the quadratic formula we have 𝑥𝑥 = −1±√-4, so there are no real solution;
F) subtracting 14𝑥𝑥 + 30 from both sides gives 𝑥𝑥2 − 24𝑥𝑥 + 140 = 0 and since 𝑥𝑥2 − 24𝑥𝑥 + 140 = (𝑥𝑥 − 10)(𝑥𝑥 − 14), the solutions are 𝑥𝑥 = 10, 14.
Question 2: write down the solutions to the following equation:
(𝑥𝑥 − 5)(𝑥𝑥 + 3)(1 − 𝑥𝑥) = 0.
Analysis of the question:
(𝑥𝑥 − 4)(𝑥𝑥 +7)(8 − 𝑥𝑥) = 0 , we conclude that 𝑥𝑥 = 4, −7, 8.