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The Algebra Calculator is a versatile online tool designed to simplify algebraic problem-solving for users of all levels. Here's how to make the most of it:

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• The golden rule of algebra states Do unto one side of the equation what you do to others. Meaning, whatever operation is being used on one side of equation, the same will be used on the other side too.
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• The basic laws of algebra are the Commutative Law For Addition, Commutative Law For Multiplication, Associative Law For Addition, Associative Law For Multiplication, and the Distributive Law.

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About solving equations

A value is said to be a root of a polynomial if ..

The largest exponent of appearing in is called the degree of . If has degree , then it is well known that there are roots, once one takes into account multiplicity. To understand what is meant by multiplicity, take, for example, . This polynomial is considered to have two roots, both equal to 3.

One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development.

Systems of linear equations are often solved using Gaussian elimination or related methods. This too is typically encountered in secondary or college math curricula. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools.

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Calculus Calculator

Get detailed solutions to your math problems with our calculus step-by-step calculator . practice your math skills and learn step by step with our math solver. check out all of our online calculators here .,  example,  solved problems,  difficult problems.

Here, we show you a step-by-step solved example of calculus. This solution was automatically generated by our smart calculator:

We can solve the integral $\int x\sin\left(2x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

 Intermediate steps

The derivative of the linear function is equal to $1$

First, identify or choose $u$ and calculate it's derivative, $du$

Now, identify $dv$ and calculate $v$

Solve the integral to find $v$

We can solve the integral $\int\sin\left(2x\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Differentiate both sides of the equation $u=2x$

Find the derivative

The derivative of the linear function times a constant, is equal to the constant

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

Isolate $dx$ in the previous equation

Substituting $u$ and $dx$ in the integral and simplify

Take the constant $\frac{1}{2}$ out of the integral

Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$

Replace $u$ with the value that we assigned to it in the beginning: $2x$

The integral of a function times a constant ($-\frac{1}{2}$) is equal to the constant times the integral of the function

Now replace the values of $u$, $du$ and $v$ in the last formula

Apply the formula: $\int\cos\left(ax\right)dx$$=\frac{1}{a}\sin\left(ax\right)+C$, where $a=2$

Simplify the expression inside the integral

The integral $\frac{1}{2}\int\cos\left(2x\right)dx$ results in: $\frac{1}{4}\sin\left(2x\right)$

Gather the results of all integrals

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

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