logic problem solver calculator

A Logic Calculator

Logic Calculator: Truth Tables

(featuring a purple monster and a psychic duck)

Introduction

This truth-table calculator for classical logic shows, well, truth-tables for propositions of classical logic. Featuring a purple munster and a duck, and optionally showing intermediate results, it is one of the better instances of its kind.

Use the buttons below (or your keyboard) to enter a proposition, then gently touch the duck to have it calculate the truth-table for you. Each time you touch the friendly monster to the duck's left, it will eat up a character (or, if there is selected text, the whole selection). If you double-click the monster, it will eat up the whole input (yum!).

Once the table is there, use the button "Show intermediate results" or "Hide intermediate results" to show or hide intermediate results in the table.

Since 2021 you may enter more than one proposition at a time, separating them with commas (e.g. " P∧Q, P∨Q, P→Q"). This makes it easier e.g. to compare propositions and to check if an argument is semantically valid.

See a few examples below. Clicking on an example will copy it to the input field. Alternatively, you may leave the input field completely empty. On touching the duck, its psychic personality will find out what proposition you are thinking of (this is an experimental feature).

(P→Q) — (P∧Q) — (P∨Q) — (P⊽Q) — (P↮Q) — (P∨¬P) — (P∧¬P) — (P→((Q→R)∧(¬S∨R))) — (P→Q)∨(Q→R) ((P ⊼ Q) ⊽ (P ⊼ Q))

If you prefer using your keyboard, you may use the strings "&" or "&&" to denote "∧"; the strings "|" or "||" to denote "∨"; the strings "=>" or "->" to denote "→"; the string "<=>" or "<->" to denote "↔"; and the strings "!" or "~" to denote "¬".

The tables are calculated in your browser, so the calculator is available offline, and the government won't ever find out what propositions you are working with (unless it hacks your computer).

Proposition

~!,]+" style="width:97%;">

No errors so far.

Truth Table

Show intermediate results

You haven't entered a proposition yet.

Operating the Logic server currently costs about 113.88€ per year (virtual server 85.07€, domain fee 28.80€); hence the Paypal donation link.

LogicalSolver

LogicalSolver supports you in solving a logic grid puzzle that is also known as logical , logigram or logiquiz . Read the Help .

To use LogicalSolver, you have to enable JavaScript.

• Click left for a negative relation (-), click right (or long-click, or while Ctrl key pressed) for a positive relation (+).
• To undo/redo, go backward/forward in your browser.
• Bookmark/share the link for saving/sharing the current state of the puzzle, including item names and entered relations.
• There is also an offline version available. Just unpack and open the contained logicalsolver.html in a browser.
• Einstein's famous Zebra puzzle can be preconfigured .
• The entries for the Jindosh riddle from Dishonored 2 can be preconfigured in English , German , French , or Japanese .

Interested in physics? Check out WaveWorkshop !

• Math Calculator

Discrete Mathematics Calculator

Solve discrete mathematics problems step by step.

The Discrete Mathematics Calculator is designed to help you quickly solve discrete math problems. Whether you're dealing with combinations and permutations, boolean algebra, or creating truth tables, our calculator has got you covered.

How to use the Discrete Mathematics Calculator?

Select a calculator.

Start by choosing the specific calculator that matches the discrete mathematics problem you need to solve. We offer calculators for combinations and permutations, truth tables, and boolean algebra.

Once you've selected the appropriate calculator, you'll typically see input fields or boxes where you can enter your data. Make sure to input the information accurately to get the correct results.

Calculation

After entering all the available data, click the "Calculate" button. The calculator will process the input and provide the solution to your discrete mathematics problem.

Depending on the calculator, you may receive a single answer, a series of values, or a truth table. Carefully study the output to understand the solution to your problem.

What Does Discrete Mathematics Mean?

Discrete Mathematics is a branch of mathematics that studies discrete (as opposed to continuous) objects and structures. This means that it focuses on things that can be clearly divided and counted, rather than things that flow or change smoothly.

Applications of discrete mathematics are vast and can be found in computer science, statistics, operations research, and many other fields. Because of its applicability to real-world problems, especially in computer science, discrete mathematics is critical for anyone interested in algorithms and data structures.

What Calculators Does eMathHelp Offer?

Combinations and permutations calculator.

• Concept: Combinatorics is a branch of discrete mathematics that involves counting, arranging, and selecting objects. This calculator assists in calculating combinations and permutations, which are fundamental in various scenarios, including combinatorics and probability problems.
• Permutations without repetitions: $$$ P(n,r)=\frac{n!}{(n - r)!} $$$
• Combinations without repetitions: $$$ C(n, r)=\frac{n!}{r!(n - r)!} $$$
• Example: Suppose you have 5 different books and you want to find the number of ways to arrange 3 of them on a shelf. You would use the permutations formula to find that there are $$$ P(5, 3)=60 $$$ ways to arrange the books.

Truth Table Calculator

• Concept: Truth tables are used in logic to display all possible combinations of inputs and the resulting outputs for a given boolean expression. This calculator generates truth tables, helping you analyze the behavior of logical circuits and expressions.
• Formulas: There isn't a specific formula for truth tables, but they are generated based on the logical expressions provided.
• Example: Suppose you have a logical expression $$$ a\wedge\left(b\vee\neg c\right) $$$ . The truth table calculator will create a table showing all possible combinations of $$$ a $$$ , $$$ b $$$ , and $$$ c $$$ , as well as the result of the expression for each combination.

Boolean Algebra Calculator

• Concept: Boolean algebra is a branch of mathematics that deals with binary variables and logical operations. This calculator allows you to work with boolean expressions, perform AND, OR, NOT, and XOR operations, and simplify complex boolean expressions.
• AND operation: $$$ A\wedge B $$$
• OR operation: $$$ A\vee B $$$
• NOT operation: $$$ \neg A $$$
• XOR operation: $$$ A\oplus B $$$
• Boolean expression simplification rules, such as De Morgan's laws and distribution laws.
• Example: You have the boolean expression $$$ \left(A\wedge B\right)\vee\left(A\wedge C\right) $$$ . You can use the calculator to simplify it to $$$ A\wedge\left(B\vee C\right) $$$ for easier analysis.

What is the Discrete Mathematics Calculator?

The Discrete Mathematics Calculator is designed to assist with various discrete mathematics tasks. It's particularly useful for calculating combinations and permutations, creating truth tables, and simplifying boolean expressions.

Why would I need the discrete math calculator?

Discrete mathematics involves complex calculations and logical operations. Our calculator helps with them, making solving problems related to combinatorics, boolean algebra, and logic circuits easier.

What are some common applications of the Combinations and Permutations Calculator?

You can use the Combinations and Permutations Calculator to quickly find the number of ways to arrange items, choose subsets, and solve combinatorics problems in probability, statistics, and more.

Is this the best calculator for discrete math?

Our Discrete Mathematics Calculator is designed to be one of the best resources for discrete math calculations. It offers a range of tools to handle various discrete math problems.

• Predicate logic
• Propositional logic

Solve a predicate logic reasoning task: help

What is this, basic syntax, what does it mean, special symbols and additional constructions.

Symbols in double quotes like in "person" stand for for distinct symbols which can be viewed as a string type . A distinct symbol is not equal to any other syntactically different symbol and not equal to any numbers or lists.

Automated reasoning

• Automated Theorem Proving Course Content by Geoff Sutcliffe is probably the best introductory material
• Handbook of Practical Logic and Automated Reasoning contains code examples in ML in addition to an excellent text
• Handbook of Automated_Reasoning for in-depth papers ( here on Amazon ).
• Online solvers on TPTP
• The CADE ATP System Competition
• SMT (satisfiability modulo theories) solvers used for formal verification like Z3 and PVS
• Answer set based solvers like DLV and Smodels , see the competition
• Description logic solvers, see the list of implementations
• OWL (web ontology language) solvers used in the context of semantic web , see the list of implementations

What is the practical use?

Prove a predicate logic theorem, advanced use, example 1 for basics.

This is a really trivial example.

We will give two facts: john is a father of pete and pete is a father of mark . We will ask whether from these two facts we can derive that john is a father of pete : obviously we can.

The facts and the question are written in predicate logic, with the question posed as a negation, from which gkc derives contradiction.

Why the negation and contradiction? This is just a convenient way to simplify the problem. Here we really want to prove a & b => a . This is true if and only if -(a & b => a) is false. Now, the last formula is equivalent to a & b & -a . Thus the input facts and rules stay as they are, and we only negate the conclusion to be proved. To summarize, giving a goal to be proved from axioms (i.e. known facts / rules) as a negated statement is just a convenient way to organize proof search and there is nothing really special about it.

Notice that each statement in the input file is terminated with a period sign. The list of statements is assumed to be a conjunction ( and ) of all statements. Minus sign - means negation. Thus -father(john,pete) means john is NOT the father of pete .

Percentage character % is a line comment. Multi-line comments /* .... */ can be used as well.

The order of statements is not really important, although it may cause minor and essentially unpredictable differences in search behaviour. Essentially, gkc creates and tries out its own search strategies, like SQL databases. It does not follow a pre-determined predictable search strategy like PROLOG.

The proof block gives us numbered steps of the proof found: each step is either a used input fact / rule or a derived fact / rule.

The [in] means that this fact/rule was given in input.

The [mp, 1, 2] (not exactly present in this proof) means that this fact / rule was derived by modus ponens (i.e. the resolution rule ) from previous steps 1 and 2. More concretely, the first literals of both were cut off and the rest were glued together.

The [simp, 1, 2] means the same as the [mp,...] above, just a specially simple case.

The Resolution (logic) wiki page is a good short intro to the general principles of the derivation rules with the course materials of Geoff Sutcliffe being a suitable continuation towards deeper understanding. However, the following examples are understandable without in-depth theoretical background.

Try to modify the example by removing the statement -father(john,pete). and run the prover again. After ca one second it will stop and output

Although the task is trivial, gkc does not attempt to be clever and tries out a large number of search strategies: this is why it takes a second to terminate. You can try running gkc with the optional switch producing more information to see the whole process: click on the Advanced button and select a higher print level or set the Show derived to on

In case gkc does not understand the syntax of the input file it will give a json-formatted error indicating a culprit line and piece of input like this:

Example 2 for answers

Like example 1, but we want to find a concrete person as an answer: we use the special $ans predicate for this. Observe the answer: $ans(pete). line in the output stemming from this answer predicate.

We are using a variable here: any word starting with a capital letter is considered to be a variable. X is a variable in our example. You could also use, say Something as a variable: it also starts with a capital letter. All the words starting with a non-capital letter are constants, functions or predicates.

Vertical bar | is logical or .

Input file example2.txt:

Notice that gkc outputs a line answer: $ans(pete). indicating the substitution pete made for X is the answer we were looking for.

Modify the example and try out the line

You will get answer: $ans(john,pete). in the output. It is possible to force gkc to give more answers than just one: more about that later.

Example 3 for rules

Now we add a grandfather rule and ask for a grandchild of John.

Input file example3.txt:

Example 4 for indefinite answers

Let us make it unclear which sons pete actually has.

Input file example4.txt:

Indeed, there is no way to give a definite answer, but gkc gives a correct answer indicating that either mark or mickey is a grandson of john, or perhaps both are.

The [mp, 1.2, 2, fromaxiom] means that the 2th (numeration 0,1,2,...) literal in the clause at proof step 1 was cut off with the first (0th) literal of the clause at proof step 2. In case the first literal is cut off, the N.0 is simplified to N, as in the previous examples.

I.e. literals in a clause are numbered 0, 1, 2, etc and the number 0 is not added to the step number.

Example 5 for more rules

To make matters a bit more complicated, we add an ancestor rule and look for ancestors of mark.

Input file example5.txt:

Example 6 for equalities and functions

Now we reformulate the whole thing with equalities and functions!

father(john)=pete means, as expected, that pete is the father of john and there can be no other fathers. If you also gave father(john)=lucas this would make gkc to conclude that pete and lucas are the same object, i.e. pete=lucas.

Importantly, two different constants are not considered inequal by default (think of the constants as labels on objects: there could be several different labels on one object): a=b does not give a contradiction.

Notice that the following proof does not use equalities, just functions.

See that gkc was happily answering father(mark) although we have not said who the father of mark actually is! The functions like father do not have to be defined on all the possible objects, they can be partially known and partially unknown.

Example 7 for equality

The proof first rewrites the clause 1 with equality clauses 2 and 3 in the [simp, 1, 2, 3] step. Next it uses the reflexivity rule r= which is basically the standard property of equality: X=X .

Example 8 for multiple answers

Now let us look at how to get several answers, not just one. We will also introduce mothers and a mother-side grandfather rule.

The "max_answers":2 part tells gkc that it should try to find at least 2 answers and stop after that.

The "strategy": ["negative_pref"] indicates that one specific strategy (here a conventional negative-preference ordered binary resolution) is to be used for proof search, without any parallel or sequential attempts with different search strategies.

The [=, 1, 2.0.2, fromaxiom] means that the clause at step 1 was used to replace a 2th subterm (numbering 0,1,2,...) of the 0th atom of a clause at step 2 using equality.

Observe that the two answers have (obviously) different proofs.

Try to modify the strategy into ask for three answers: "max_answers":3, and then run gkc again. It will give the same output as before, but will add the last line at the end:

essentially indicating that the required number of different proofs (3) were not found.

Example algebra

Next we will take a question from abstract algebra, axiomatized wholly by equalities.

The proof uses given equalities to derive several new equalities. The = rule basically replaces parts of one premiss matching (unifying) one side of the second premiss equality with the other side of the equality.

The simp rule also replaces a part of a premiss, but does so by rewriting , meaning that the original unchanged premiss is not used in the search after the simplification replacement.

The [=, 1, 2.0.3, fromaxiom] line means that the clause at step 1 was used to replace a 3th subterm (numbering 0,1,2,...) of the 0th atom of a clause at step 2 using equality.

Example with an unprovable problem

A simple clause set in this example produces an unlimited number of clauses like

and obviously does not lead to a contradiction.

However, gkc does not attempt to detect unprovability and will run either forever or for a very long time, until the browser complains and asks you to stop, or it runs out of allocated time or memory.

NB! You will have to reload the page after you stop the script, otherwise the page will be unresponsive.

Example with a hard problem

It will probably run until the browser complains badly or time given by the automatic strategy selection runs out.

Of course, it is likely that with a suitable search strategy gkc will find a proof using a sensible amount of time and memory. Currently we are simply unaware of which strategy it would be!

Blocks world

Blocks world is a classic family of toy problems: there is a robot arm able to lift single blocks and to put them on top of other blocks.

The goal is to find a sequence of robot arm movements to produce a required configuration of blocks: for example, a tower. The comments in the problem text explain the initial situation and the arm movement rules.

Without a specialized search strategy these planning problems tend to be surprisingly hard for provers, including gkc.

Example 14 for

Example 13 for various characters.

This is not a really advanced topic, but important.

Gkc follows the TPTP conventions in the way it treats special characters, with a few additions. A normal symbol must not contain any whitespace or non-alphanumeric characters like (, -, ~, =, ", comma etc: however, underscore _ and dollar $ are allowed, with predicates and functions with a special predefined meaning being prefixed with a dollar $.

Any symbol containing a character except an ascii letter, digit, underscore _, or dollar $ will be printed out by surrounding it with single quotes. As an exception in gkc, equality = and conveniece infix forms of arithmetic expressions +, *, -, /, will not be surrounded by quotes.

Double quotes like "John Smith" indicate that a symbol is distinct , i.e. unequal to any other distinct symbol, number or a typed object. Double quotes inside double quoted symbols must be prefixed by a backslash. More about distinct symbols in the later examples.

Additionally gkc allows to make a symbol variable by prefixing it with a question mark like this: ?smth . Any symbol starting with a capital letter or a question mark is assumed to be a variable, and the rest are not.

This holds for both the simple syntax in the previous examples and the TPTP fof formulas to be described next.

Thus, for gkc in fof formulas a capital-letter-starting symbol is a free variable even if it is not explicitly quantified: since this could be confusing, it may be better to avoid such symbols unless they are explicitly quantified.

You can also use integers like 71 or period-separated decimals like 1.35 as constants.

Gkc is agnostic towards using different character encodings: it uses c-strings, i.e. 0-terminated byte sequences and does not care about encodings.

Example 14 for TPTP syntax with steam

See the official fof syntax and fof example in TPTP .

Gkc will first convert the complex formulas to a simple clause form (properly called clause normal form ) used in the previous examples. The statements we had in these examples are called clauses .

Each statement in the TPTP fof language is terminated with a period symbol . and has a structure

where the statement name will be used in the proof, the statement role indicates whether it is an axiom, an assumption or hypothesis, or a goal to be proved from these: the latter is either conjecture (which has to be negated) or negated_conjecture (negated already).

Indicating the role enables provers to find a more suitable strategy. It does not (except the conjecture case, which has to be negated) have a logical meaning.

The second statement is existentially quantified ( ? symbol) and will be converted by gkc to a clause wolf($sk7). where $sk7 is a new constant invented by gkc which should not occur in any other formula in the problem: this procedure is called Skolemization . Gkc always uses the $sk prefix for such constants and functions, using a new number N for each new one. The original formula is assumed not to contain $skN form symbols.

The last statement expresses the question to be proved: is there an animal that likes to eat a grain eating animal? It has a conjecture role and has to be first negated and then converted to a clause:

In general, one formula in the input may create several clauses and the optimized algorithm for creating such clauses is nontrivial, sometimes involving mini-scoping and the creation of new definitions (new predicates) to make the size and number of generated clauses smaller.

The simple format proof contains only clauses created from the formulas: the original names of formulas used are indicated like

The simple output format used in all the previous examples does not directly contain the original input formulas of the fof language of TPTP and the steps in conversion, only the names of the formulas. TPTP suggests using a particular format of output, including the original input before clausification, proved-successfully-marker, and the begin and end markers of proof.

Steam using simple formulas

By default gkc does not know which clause is a question: this is ok, but it is bad for efficiency in case of large clause sets.

There is a simple way to tell gkc that some clause or a formula is indeed a question and gkc should heavily focus on that clause or a formula.

Instead of writing your negated question like

write it with this wrapping as a fof formula (observe double parentheses at the end)

or a cnf formula (i.e. simple clause) in TPTP language

and then gkc knows that this clause should get priority in search.

Equivalently, you could pose the question in a positive form and tell gkc that it is a *conjecture*:

Similarly, you can tell gkc that a clause or a formula is not just an arbitrary axiom which may or may not be used, but an important assumption likely to be used in the proof:

Example 16 for the strategy

You can tell gkc the exact strategy by entering the json format strategy text like this into the Strategy as json text box:

There are two somewhat different ways to write the strategy file: either indicate a single strategy (single run) like in the example above or multiple runs like this:

In the latter case gkc will try out all these strategies in that order

A simple way to obtain such run sequences is to use the Print level select box value + Statistics which will then automatically construct and print out a suitable full strategy json with many runs, which you can simply copy and paste into your own file for later modification.

The default value for limit-type fields starting with max_ is 0, indicating that no limit is set.

In case "equality": N is not set to 0, GKC uses reflexivity, paramodulation and demodulation with knuth-bendix ordering for handling equality.

The list "strategy": [...] contains the main search strategy indicators, default off:

• "query_focus" : use a goal-oriented set-of-support strategy with binary resolution
• "negative_pref" : use binary resolution with negative literals preferred
• "positive_pref" : use binary resolution with positive literals preferred
• "hardness_pref" : use binary resolution with "hardest" (similar to weight) literals preferred
• "knuthbendix_pref" : use binary resolution with knuth-bendix ordering of literals
• "hyper" : use hyperresolution, with negative literals preferred
• "posunitpara": perform paramodulation from units only
• "prohibit_nested_para": disallow paramodulation if either parent is derived by paramodulation
• "max_ground_weight": use the weight of the heaviest literal as the base weight of a clause
• "unit", "double" or "triple" : use binary unit resolution or its generalization: (one of the arguments must be unit, a two-literal or three-literal clause, respectively. These may be added to the list in addition to the previous strategy indicators, for example, like ["query_focus","unit"].

Other useful parameters, to be used outside the "strategy": [...] list:

• "print": 0 or 1, where 0 prohibits almost all printing, default 1.
• "print_level": integer determining the level of output: useful values are between 0 and 50, default 10, for -tptp 1 gkc uses 15.
• "print_json": 0 or 1, where 0 is default and 1 forces json output.
• "print_tptp": 0 or 1, where 0 is default and 1 forces tptp-style proof output
• "max_size", "max_length", "max_depth", "max_weight" indicate limits on kept clauses, defaults are 0.
• "equality" : 1 or 0, with 1 being default and 0 prohibiting equality handling.
• "rewrite" : 1 or 0, with 1 being default and 0 prohibiting using equations for rewriting.
• "max_dseconds": N being an integer limit of tenths of seconds for one run, default 0 (no limit).
• "max_seconds": N being an integer limit of seconds for one run, default 0 (no limit). This is an alternative to the "max_dseconds" key.
• "weight_select_ratio": N indicating the ratio of picking by order derived / clause weight, default is 5.
• "max_answers": N indicating the maximal number of proofs searched for until search stops, default is 1.
• "reverse_clauselist": N either default 0 or 1, where 1 follows the actual order for input clauses, starting from the end.
• "sine": input filter with N either 1 or 2 where 1 is a less restrictive (accepts more clauses) and 2 more restrictive. NB! Sine is automatically switched off if the -provekb switch is used.
• "depth_penalty": additional penalty for clause depth, default 1
• "length_penalty": additional penalty for clause length, default 1
• "var_weight": weight of a variable, default 5
• "var_weight": weight of a repeated variable, default 7
• "query_preference": N being 0, 1, 2 or 3 indicates which parts of the problem are treated as goals, assumptions or axioms: 0 stands for no goal/assumption preference. 1 stands for input preference (the assumption and conjecture formulas of fof) 2 stands for making non-included formulas assumptions 3 stands for considering only the negative clauses from conjecture to be goals

For "max_seconds"<2 gkc will automatically use immediate check for contradiction when a clause is derived.

Example 17 for arithmetic

Gkc supports integers and floating point numbers along with basic arithmetic operators, but does not provide any axioms or built-in knowledge for the properties of arithmetic operators, except simple evaluation.

As a trivial practical example gkc is unable to find that

is contradictory. This would require either solving the equation 5=2+X or generating an ever-growing set of numberic instances of clauses, none of which gkc currently attempts.

However, for the last example one can construct a number-generating clause set - which essentially enforces the creation of numeric instances - like this:

leading to a proof.

Example 18 for lists

The current example defines functions for counting the length of the list and summing the numeric elements. Then it uses these functions in a rule for deriving a goodcompany type as an object with at least three customers and revenues over 100. Finally we ask to find an object with the goodcompany type.

Example 19 for distinct symbols

Gkc follows the TPTP syntax of using double quotes like in "person" for distinct symbols which can be viewed as a string type . A distinct symbol is not equal to any other syntactically different symbol and not equal to any numbers or lists.

Example 20 from natural language

Example 21 from organization theory, example 22 from boolean algebra, example 23 from medicine, example 24: dreadbury mansion, example 25 from group theory, example 26 from set theory.

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What Is the Truth Tables Calculator?

Boolean algebra, applications of boolean algebra, boolean logic of primary operations, solving for truth tables.

How to Use the Truth Tables Calculator?

How does the truth tables calculator work, solved examples, improper integral calculator < math calculators list > simpson’s rule calculator.

• Mathematical logic signs and operations

How to use it?

• Expression :
• ﹁(1&1V0)↔(﹁1V1)
• ¬((P∨¬Q)∧¬R)
• (1v0)&(0&a)v(1v0)

Mathematical logic step by step

The teacher will be very surprised to see your correct solution 😉

• Use symbolic logic and logic algebra
• Place brackets in expressions, given the priority of operations
• Simplify logical expressions
• Build a truth table for the formulas entered
• Conjunctive normal form (CNF), including perfect
• Disjunctive normal form (DNF), including perfect

Examples of logical expressions

• Using negative
• (A⇒B)⇒¬(B⇒A)
• Addition of logical expressions
• (A⊕B)∨(A⊕C)
• With Equivalent Sign
• (A⇒B)∨(B⇔C)
• With "Consequence" Sign
• ((A⇒B)⇒(A|C))⇒(¬B⇒¬C)
• (A⇒B)∧(A↓C)
• With the use of conjunction and disjunction
• (A∨B)∨C ⇒ (A∨B)∧(A∨C)
• With the use of Not-And and Not-Or
• 0↓1|a|b|c|1↓0

Supported symbols in logical expressions

Here are the symbols that should be specified when entering a logical formula into the calculator

In the calculator, you can simplify expressions with the following operations: NOT, XOR, AND, OR, NAND, NOR, NOT, XNOR

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Number line.

• x^4-5x^2+4=0
• \sqrt{x-1}-x=-7
• \left|3x+1\right|=4
• \log _2(x+1)=\log _3(27)
• 3^x=9^{x+5}
• What is the completing square method?
• Completing the square method is a technique for find the solutions of a quadratic equation of the form ax^2 + bx + c = 0. This method involves completing the square of the quadratic expression to the form (x + d)^2 = e, where d and e are constants.
• What is the golden rule for solving equations?
• The golden rule for solving equations is to keep both sides of the equation balanced so that they are always equal.
• How do you simplify equations?
• To simplify equations, combine like terms, remove parethesis, use the order of operations.
• How do you solve linear equations?
• To solve a linear equation, get the variable on one side of the equation by using inverse operations.

equation-calculator

• High School Math Solutions – Quadratic Equations Calculator, Part 1 A quadratic equation is a second degree polynomial having the general form ax^2 + bx + c = 0, where a, b, and c...

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