A quadratic equation is a specific type of equation, different than the linear equations we’ve worked with before. To be more specific, a quadratic equation is any equation that can be written in the form: ax2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. (If a = 0, the equation is a linear equation.)
Solving quadratic equations is a high emphasis topic on the TASC Test Assessing Secondary Completion™ Mathematics subtest. Though solving a problem like this may seem complex, specific tools can help you work through any quadratic equation as you prepare for the Math subtest. In fact, there are four options for solving quadratic equations in one variable. Depending on how much information – and what information – is provided within the equation, you can take any of these approaches.
Here, we’ve outlined how to factor quadratic equations. We chose to focus on factoring because it’s typically understood to be the easiest way to solve a quadratic equation. It’s also one of the most common methods.
To factor, you must follow these steps:
- Write your Equation in the form ax2 + bx + c = 0. You can do this by rearranging the formula, as we discussed in our earlier post. You’ll remember this can mean a number of things. You might have to use the distributive property, combine like terms, move terms to different sides of the equation, or apply the order of operations.
- Factor your equation.
- Let each factor equal 0, and solve.
Walk through solving one example of a quadratic equation:
(2x + 3)x = 5
First, notice the x is against the parenthesis. This means that it is being multiplied and you have to distribute it.
2x2 + 3x = 5
Then, you want to have all of your numbers on one side of the equation, to make sure it is in the right form.
2x2 + 3x = 5
(2x2 + 3x) – 5 = (5) – 5
2x2 + 3x – 5 = 0
Now that it’s in the right form, you can factor it.
(2x + ?)(x – ?) = 0
Here, the two unknown numbers (?) must multiple to equal (-5) when you use the Distributive Property to multiply out. The first two terms must multiply to equal 2x2, and the middle products must equal 3x. Therefore, the answer must be:
(2x + 5)(x – 1) = 0
because when factored out we get the original:
2x2 – 2x + 5x – 5 = 0
2x2 + 3x – 5 = 0
Using our complete factors, we can apply the Zero Product Law, which allows us to state:
(2x + 5) = 0
(x – 1) = 0
To solve the quadratic equation, we can solve these two equations:
(2x + 5) – 5 = 0
2x = -5
(2x)/2 = (-5)/2
x = (-5)/2
(x – 1) + 1 = 0
x = 1
Therefore, we have found the roots of the quadratic equation and can graph the results. You may only be asked for the roots on the math subtest, or you may be asked to identify what the graph looks like (in which you’ll need to find the roots to identify where the line passes through the plots). It is helpful to note that quadratic equations always make nice curves.
The sources we have included here also describe the other methods for solving quadratic equations, including: Extracting Square Roots, Completing the Square, and Using the Quadratic Formula. Have questions about factoring or any of these additional methods? Leave a comment below and we’ll help you!
A citation refers to a moment in a piece of writing when you, as the writer, note that something you have written comes from another source. Citations serve two purposes: most importantly, they add weight to your argument – if a source has a certain level of authority or expertise, quoting that source adds to your authority. Secondly, citations enable your reader to follow your train of thought and trace back your argument.
Different citation styles serve specific purposes. They were created, and are regulated, by institutions that govern types of research. For example, two of the citation styles discussed below were established by the primary institutes in their respective fields: MLA by the Modern Language Association, and APA by the American Psychological Association.
Some of the best writing guidance we can give you is how to use citations on your TASC Test Assessing Secondary Completion™ Writing subtest. Being able to conform to the guidelines in a style manual that is appropriate for your discipline and writing type is a high emphasis topic. We introduce three types of citation styles below, and give you some tips regarding when you should cite – and the difference between quoting and using your own words. Navigating this distinction can improve your writing and your overall argument.
MLA is traditionally used for writers working in the Humanities, in disciplines such as Art and English Literature. When citing in MLA, the emphasis in your citation is on the author’s name rather than on the year the material was published (regardless of how well-known the author is).
Additionally, in writing, the emphasis is on “unpacking” what is important. By unpacking, we mean explaining and interrogating why a given quotation from a source is important and worth quoting. This is sometimes referred to as close reading.
In more formal writing, you would have a Works Cited page that collects all of the information about the sources included throughout your paper. On the TASC Writing subtest, however, you only need to include in-text citations. For that reason, we’ve outlined a few types of in-text citations that may be helpful to you:
- If you are referring to a quotation from a passage that has an author or organization indicated, format your parenthetical citation this way:
(Last Name Page Number)
- If there is no author, you’ll format it this way:
(“Title of Passage” Page Number)
(“A Rose for Emily” 2)
- If there is no page or line number, only use the author’s name.
Always place in-text citations in parenthesis at the end of a sentence. Remember, you can only work with the information provided with the passage on the TASC Writing subtest. Be sure to note the author’s name, the publication year, and any other additional information that can help you clarify your in-text citation.
Unlike MLA, APA was designed for writers working in the Sciences and the Social Sciences. The emphasis in this style is on the year the material was published. This emphasis is primarily because in these fields the most recent research is the most important, and older research is outdated.
In writing, quotations are typically used to introduce or to reinforce data. It is less about close reading, and more about providing essential information that supports the writer’s hypothesis.
The general style is similar to MLA. You will still use parentheses for your in-text citation at the end of your sentence. Additionally, you would include a Works Cited page at the end of a more formal writing assignment. For the TASC Writing subtest, however, you’ll want to use only in-text citations from any quotations you pull from the passage you’ve read and are responding to. Here are the same styles we presented in MLA, in APA format:
- For a passage with an author or organization:
(Author, Date, Page Number)
(Dickens, 1859, 101)
- If there is no author:
(“Title of Passage,” Date, Page Number)
(“A Rose for Emily,” 1930, 2)
- If there is no date:
(Author, Page Number)
- Like MLA, if there is no page number, only include the author and date.
When to Cite
You cannot cite everything, and when you are reading a passage on the writing subtest, be selective about what you include in your response. If a quote meets any of these criteria, we think it’s a good rule of thumb to include it:
- Language is especially vivid or expressive
- Exact wording is necessary
- Debater’s voice is important
- Authority lends weight to your argument
- Analyzing or interpreting
If not, you can probably paraphrase or summarize the information. Paraphrasing refers to discussing something that was written about in a passage, in your own words. If you can say it just as well – or more concisely – than the author of the passage, you should paraphrase. If there is a page number, you should still include that in parentheses after the paraphrase.
The above citations are just a few examples of the rules for citing for these different styles. Websites like the Online Writing Lab (OWL) at Purdue, Diana Hacker’s Style Guide, and Turabian’s Manual for Writers can help you navigate more nuanced citations, such as online periodicals or media. If you are planning on pursuing a post-secondary degree after completing your high school equivalency, these websites are helpful for writing research papers.
Writers use figurative language for descriptive effect, according to Kayla Nabholz of Arkansas Tech University. It is not used in a strictly literal sense. In fact, as you might remember from our earlier post A Guide to Meaning: Figurative, Connotative, & Technical | TASC
Reading, the opposite of figurative language is literal language. As Nabholz notes, “Many common, everyday expressions are figurative, and when used imaginatively, this language can add a special dimension of meaning to both speech and writing.”
You probably use figurative language on a regular basis. Phrases like “I love you like crazy” and “You are green with envy” are figurative. Sometimes, you may hear such phrases described as metaphorical language – but metaphor is only one type of figurative language.
We discussed metaphor and simile in our previous post. However, we want to give you more information about figurative language because it is an essential component of writing. As you prepare to take the TASC Test Assessing Secondary Completion™ Reading subtest, you can improve your reading skills and better understand context by learning about figurative language. Figurative language can help you understand an author’s writing in a more complex way.
Once you have familiarized yourself with metaphor and simile, start to work with personification. When an author uses personification, he or she is giving an animal, object, or idea a human form or characteristic. An easy way to remember this figure of speech is to know that the writing describes something as being “person-like.”
“When it comes, the landscape listens,
Shadows hold their breath”
Hyperbole is a type of exaggeration, and it is usually extreme. It may be helpful to know that the word hyperbole comes from the Greek, meaning “over-casting.” This type of figurative language attracts attention, especially in spoken language. It can also be used to emphasize a point or create an effect.
You often use hyperbole when you say things like “I’m too tired to function” or “I’ve told you a million times.” Literary examples include:
“I had to wait in the station for ten days—an eternity.”
Joseph Conrad, The Heart of Darkness
“I’ll love you, dear, I’ll love you
Till China and Africa meet,
And the river jumps over the mountain
And the salmon sing in the street,
I’ll love you till the ocean
Is folded and hung up to dry”
- H. Auden
This literary device can be more complex to find in a passage than the other types of figurative language. According to the educators on the English Language Learners’ site Really Learn English, irony is “when someone says or does something, but means another thing or intends for something else to happen.” It may also be that a situation may develop in a different way than the reader anticipated. You can remember irony as simply the difference between appearance and reality.
There are three types of irony:
- Verbal irony
- Dramatic irony
- Situational irony
An author may convey irony through dialogue. As a reader, pay close attention to the characters and their interactions to decide whether or not they are being sincere within the reading.
Energy is an essential component of physics, which aligns with Forces and Interactions and can further your understanding of matter.
You might remember that physics-related science topics on the TASC Test Assessing Secondary Completion™ are medium emphasis topics. Science is often interrelated. Discoveries build on previous discoveries. Having a thorough understanding of all potential topics – both high emphasis and medium emphasis – can improve your ability to discuss scientific topics and prepare you for the Science subtest.
What is Energy?
Energy is simple: it is the ability or power to do work. It’s similar to what you need to do work. Work is the force used to move something. For example, if you need to move many heavy boxes early in the morning, you should eat a good breakfast so your body can create the energy it needs to do that work.
According to the Boston University Physics department, these two principles give us more tools to analyze physical situations. Whenever a force is applied to an object, and causes the object to move, work is done by the force. If a force is applied and the object doesn’t move, no work is done. And, if a force is applied and an object moves in a direction other than the direction of the force, then less work is done than if the object were to have moved in the same direction of the force.
You can determine either the amount of energy used or the amount of work being done by using the right formulas. First, you’ll need to know what type of energy you’re dealing with.
Types of Energy
There are two basic types of energy:
- Kinetic: This type of energy is created from a moving object.
- Potential: This type of energy is waiting to be used. You might also hear it referred to as stored energy.
Potential and kinetic energy can come in these forms:
- Heat Energy: Created by moving molecules that are either hot and are finding cold, or are cold finding heat.
- Light Energy: Created by light.
- Sound Energy: Created by the traveling vibrations we hear.
- Electrical Energy: Created by moving electrons plugged into an electrical source. Most electrical energy can be converted into another form of electricity.
- Mechanical Energy: Created by the motion of machines.
- Chemical Energy: Created by putting chemicals in contact with one another.
These forms of energy come from a variety of sources, including natural and manmade sources. You’re probably familiar with natural sources of energy – like energy from the sun, water sources, and wind. Additionally, we are able to get energy by burning wood, using geothermal steam, and working with nuclear sources.
An Important Formula
We want to highlight an important formula that will help you with energy. Though you may not be asked a question regarding these formulas or regarding energy on the TASC test, these formulas can further your understanding of the relationship between energy, work, and force. And you can use these two formulas to practice rearranging formulas for the TASC Mathematics subtest!
This important formula is the formula for kinetic energy. An object has kinetic energy whenever it has mass and it is moving. Or, as we stated above, kinetic energy is the energy of a moving object. To find kinetic energy, you would use this formula:
KE = 1/2mv2
Here, m stands for mass and v stands for speed. Remember that you can rearrange this equation to use kinetic energy to solve for mass, if speed is known, or speed, if mass is known. We recommend practicing your understanding of this equation, and being sure that you completely understand the connections between energy, work, and force.
Are you a resourceful mathematician?
Resourceful mathematicians know a variety of tips and tricks for working through different types of math problems. You’ve probably learned a few tips from the TASC Test Assessing Secondary Completion™ blog, but the best tip by far is this: formulas can be rearranged.
When you know that formulas aren’t fixed – and can manipulate them to solve for the variables that you need – you know how to make a formula work for you.
A formula is a special type of equation that shows the relationship between different variables. Formulas are similar to equations, and can be treated the same way. You’ll remember that a variable is a symbol like x. A variable stands for a number that isn’t known but typically can be solved for.
To rearrange a formula, you must first know what the subject of your formula is. For example, in the SUVAT formula (used for finding the velocity of an object over time), V is the subject of the formula:
V = u + at
Here, V is final velocity, u is initial velocity, a is acceleration, and t is time. But what if a question on the TASC test Mathematics portion asks you to solve for acceleration, rather than final velocity? You have to rearrange the formula by isolating a, or in other words making a the only letter on one side of the equation:
(V) – u = (u + at) – u
V – u = at
(V – u)/t = (at)t
(V – u)/t = a
By subtracting u and dividing by t, we successfully isolated a. Therefore, we know that the product of final velocity subtracted by initial velocity divided by time equals acceleration.
Note that in our example, anything we did to one side was also done to the other side. This keeps the formula balanced. You would do the same if you were working with an equation. For example:
x + 2 = 42
x + 2 = 16
(x + 2) – 2 = (16) – 2
x = 14
We solve the exponent first, because that step comes first in the Order of Operations. (Which, you might have been taught as PEMDAS in middle school. PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction – the Order of Operations.)
The Order of Operations can help you rearrange formulas. Whenever you use an operation to move a section of a formula, you must perform the opposite operation on the other side of the formula. In the above formula, for example, we divided t from at because a and t were being multiplied. The Order of Operations pairs those opposites together – Multiplication and Division, Addition and Subtraction – making it easy for you to remember what you need to do to make the formula work for you.
Resourceful, right? Test your skills with this worksheet from the Aquinas College Math Department.
Language is not a static thing. Like the round characters you read about, language continues to change and evolve over time. It is dynamic.
Keep this in mind as you prepare for the TASC Test Assessing Secondary Completion™ Writing subtest. Grammar conventions and usage are important to writing a successful response.
One of the Writing subtest high emphasis topics asks students to apply the understanding that usage is a matter of convention, can change over time, and is sometimes contested. But why is it important to know that language changes? And, how can it help you on the TASC test?
Language and Grammar
Grammar, as defined by the instructors at State University of New York (SUNY), is “the set of underlying rules that makes possible meaningful utterances” in any language. Grammar differs between languages and must be understood by the individuals who are using that particular language. Without an understanding of grammar, communication would be impossible.
You could say that grammar and language have a causal relationship—when grammar changes, language must change as a result, and when language changes, so must grammar. They depend on one another.
You might feel overwhelmed by English grammar to accurately understand its conventions and changes. However, as the SUNY instructors point out, if you speak the language competently, if other people can understand and communicate with you, then you actually know English grammar. You are using nouns, verbs, modifiers, and other forms of grammar, even if you do not understand the technical definitions of those terms.
Usage and How It Changes
Usage, then, is the habitual or customary practices in spoken or written language. There are some differences between spoken and written language; some usages are more appropriate for spoken language, and would be considered informal in written language. The instructors at SUNY provide a good example: “Anyone who speaks English competently understands, and knows how to use, the word ain’t, a word that has been part of the language for nearly two hundred years. Yet, at least in written English, ain’t is widely frowned upon as acceptable usage.”
Usage helps you understand how language works, and how individuals will use language in different ways to express themselves. Conventional usage might seem subjective, or even illogical. Remember that any changes made to language or grammar are in response to a larger trend of communication. When many people begin to use a word in a particular way, that use spreads and the word’s new usage becomes conventional.
Who? Why? How?
You might be left wondering who controls these changes, why it’s important, and how you can use this new information on the TASC test.
Teachers, editors, and compilers of dictionaries and usage manuals primarily determine which usages are considered acceptable or unacceptable. These educators and writers are largely responsible for regulating conventions because they are the ones who tell you whether or not your word choice is accurate and logical. Often, they take into consideration whether the word has character and if it is lucid, simple, direct, and aesthetic.
It should be noted that some conventional usages may not be considered acceptable, but may persist and be used conversationally or informally – think of slang words, such as basic or bae.
It is important for you to be aware of changes in conventions for two reasons:
- You might be reading older material while preparing for the TASC test or while taking the test. As a reader, you should always be aware of the year the work was published. The year is typically included in the title information. When it is published, and any historical context you can gain from that time period, can be helpful when it comes to understanding unfamiliar words or words that look familiar but are being used in a different context. For example, prior to the 1920s the word gay meant happy or If you are reading an excerpt from a book written in the 1800s that uses the term gay to describe one of its characters, the meaning is not the same as it is today. Use the context to help you determine changes in conventional usage like this.
- Students should be prepared to “explore how conventions are used in specific contexts and genres to achieve a particular effect with an audience,” according to Kathleen Cali and Kim Bowen of Learn NC. Test takers will be expected to understand how conventions contribute to the reader’s understanding of a text as a whole, and how an author uses those conventions to mold that understanding.
Some professional writers actually defy these conventions to achieve certain effects and impact their readers in specific ways. If you are writing a response to a specific passage, knowing how an author is using conventions can dramatically impact your reading of the passage.
How to Earn Your HSE With Your New Year Resolutions
If you’ve been thinking about earning your high school equivalency (HSE) – because you want to go back to school or you want to be more competitive in the job market – the 2015 New Year can be your perfect opportunity to set that goal. It’s a time for new beginnings, and earning your high school equivalency is your fresh start.
Resolutions can be tricky. Psychologist Jeremy Dean, writing for The Wall Street Journal, notes that 60 percent of people typically give up their resolutions after six months. He explains that this is because the typical targets for resolutions “are very difficult habits to alter because these patterns of behavior have been built up over many years.”
If you know that achieving your resolution will be difficult, you can better prepare yourself. Dean suggests you start by identifying your habits. Any habit that could conflict with your resolution to study could make it difficult for you to establish a new routine. You can replace your existing habits with better ones.
If you enjoy spending time surfing the Internet, devote some of that time to using the TASC Test Assessing Secondary Completion™ blog for exploring parts of the test and studying. If you like to read magazines before bed, try to read some of the test prep materials instead.
Making these little changes can really help in the long run. You can also make studying for your high school equivalency a part of your resolutions list with a few simple steps:
- Find a TASC test center near you. The administrators can give you insight into the test, and could help you set short-term and long-term goals. Some test centers also offer classes. One way to achieve your resolution is to sign up for classes. Even if the classes are a few weeks or even a couple months away, having these deadlines ahead of you can keep you on track.
- Buy a day planner. Your day-to-day schedule is probably busy, and when you have a lot going on at work and home, you might delay studying. Resolutions can quickly fall to the bottom of your to-do list. According to the experts at Campus Explorer, you should invest in a good day planner that will “allow you enough room to write down weekly and monthly activities, in addition to babysitter information, addresses and a master seasonal calendar that helps you map out activities by the hour.” Having a print planner can help you see everything at a glance. It can help you budget your time and make you more productive.
- Make time for your family and yourself. Having new resolutions for the New Year doesn’t mean you can’t keep some of your usual activities. If you usually have date night on Fridays with your significant other or family dinners on Sunday, you shouldn’t replace these traditions with studying. Block out free time for yourself on your schedule to give yourself breaks. It might feel like wasted time at first, but taking time for yourself can make you more productive – and, if you schedule your free time for the end of the week, it can feel like a great reward after a long week of keeping to your schedule.
- Look for other students in your area. If you feel comfortable reaching out to other high school equivalency students, you can create a study group. Even if you meet with only one other student for coffee once per month, these meetings can be a great opportunity to share ideas and review subject areas. It can also be a nice way to be held accountable.
Looking for some additional advice or have questions about planning for your high school equivalency? Leave a comment below, and someone from the TASC test team will get back to you.
Happy New Year!
Have you been studying vocabulary? The answer is probably. Whether you’ve been memorizing new terms or just looking up unfamiliar words while you’ve been reading the practice passages for the TASC Test Assessing Secondary Completion™, you’ve been expanding your vocabulary just by preparing for your high school equivalency (HSE) test.
However, vocabulary isn’t the only thing the TASC Reading subtest is testing. Unlike older tests, you won’t be asked to complete analogies. When it comes to defining unfamiliar words – and to making conclusions about the passages – you should practice your contextual skills. It’s more helpful to know how to define a word in context rather than simply memorizing a long list of words.
By context, we mean the other words and sentences that are around the unfamiliar word. As the educators at Cuesta College note, “when you figure out the meaning of a word from context, you are making a guess about what the word means. To do this, you use the hints and clues of the other words and sentences. You won’t always be right, but many times you will be.” In other words, you can get pretty close to a new word’s definition just by using the passages that are available to you on the test.
Teachers commonly encourage their students to focus on specific types of context clues, including:
- Definitions: A word or phrase typically placed in a clause immediately after the unfamiliar term, which provides an explanation.
- Synonyms: Words around an unfamiliar term that mean the same or nearly the same as the word; similar to definitions, but usually a single word rather than a phrase or clause.
- Antonyms: Words around an unfamiliar term that mean the opposite or nearly the opposite of the word.
- Examples: Often given in the following sentence and directly elaborate on the previous information in general and the unfamiliar term in particular.
- Knowledge of Subject: Prior knowledge of the topic discussed helps you understand the unfamiliar term.
Use this quick strategy to identify any of these specific types of context clues:
- If you are using a paper-based test (or you’re reading a printed article or book), you should first circle the word or phrase you don’t understand. Use scratch paper when using a computer-based test.
- Underline key phrases and ideas in the sentence, and the main idea of the paragraph.
- Look for definitions, synonyms, antonyms, or examples in the sentence and surrounding sentences.
- Try to put the sentence in your own words. If there are no synonyms available, this can help you find your own synonyms to understand the term.
- Reread the sentence replacing the unfamiliar term replaced with your synonym to see if the sentence still makes sense.
If the steps don’t work well for you, you can also ask yourself questions. Judy Zorfass and Tracy Gray of Reading Rockets suggest focusing your questions on the unknown word and the possible clues to its meaning:
- What are the surrounding words?
- How do these offer me clues?
- What does this word mean in terms of the context?
When you have the context, these questions can be much more specific. For example, if you are reading about birds and you see the word ornithology, you can ask yourself: How does the information I see here about birds help me understand this word? (Ornithology is the study of birds.)
Apply these steps while you are reading, and you will quickly get into the habit of using context clues to further your understanding. You should practice this whenever you read – not just when you are practicing for the TASC test. Though it’s a great reading study tip, it’s also a great habit to have. The more you practice, the more habitual it will be.