How to solve cubic equations. How to solve cubic equations Rules apply to a server-based calculator

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There are additional
materials in Special Section 555.
For those who are "not very ..."
And for those who "very much ...")

What "square inequality"? No question!) If you take any quadratic equation and replace the sign in it "=" (equal) to any inequality icon ( > ≥ < ≤ ≠ ), we get a square inequality. For instance:

1. x 2 -8x + 12 ≥ 0

2. -x 2 + 3x > 0

3. x 2 ≤ 4

Well, you get the idea ...)

It is not for nothing that I have linked equations and inequalities here. The point is that the first step in solving any square inequality - solve the equation from which this inequality is made. For this reason, the inability to solve quadratic equations automatically leads to a complete failure in inequalities. Is the hint clear?) If anything, see how to solve any quadratic equations. Everything is detailed there. And in this lesson we will deal with inequalities.

The inequality ready for solution has the form: left - square trinomial ax 2 + bx + c, on the right - zero. The inequality sign can be absolutely any. First two examples here already ready for a solution. The third example still needs to be prepared.

If you like this site ...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

In a cubic equation, the highest exponent is 3, such an equation has 3 roots (solutions) and it has the form. Some cubic equations are not so easy to solve, but if you apply the correct method (with good theoretical training), you can find the roots of even the most complex cubic equation - for this use the formula for solving the quadratic equation, find the whole roots, or calculate the discriminant.

Steps

How to solve a cubic equation without a free term

Find out if the cubic equation has a free term d (\\ displaystyle d) . The cubic equation has the form a x 3 + b x 2 + c x + d \u003d 0 (\\ displaystyle ax ^ (3) + bx ^ (2) + cx + d \u003d 0)... For an equation to be considered cubic, it is sufficient that only the term x 3 (\\ displaystyle x ^ (3)) (that is, there may be no other members at all).

Take out the brackets x (\\ displaystyle x) . Since there is no free term in the equation, each term in the equation includes the variable x (\\ displaystyle x)... This means that one x (\\ displaystyle x) can be excluded from parentheses to simplify the equation. Thus, the equation will be written like this: x (a x 2 + b x + c) (\\ displaystyle x (ax ^ (2) + bx + c)).

Factor (the product of two binomials) the quadratic equation (if possible). Many quadratic equations of the form a x 2 + b x + c \u003d 0 (\\ displaystyle ax ^ (2) + bx + c \u003d 0) can be factorized. Such an equation will turn out if we take out x (\\ displaystyle x) outside the brackets. In our example:

Solve a quadratic equation using a special formula. Do this if the quadratic equation cannot be factorized. To find two roots of an equation, the values \u200b\u200bof the coefficients a (\\ displaystyle a), b (\\ displaystyle b), c (\\ displaystyle c) substitute in the formula.

• In our example, substitute the values \u200b\u200bof the coefficients a (\\ displaystyle a), b (\\ displaystyle b), c (\\ displaystyle c) ( 3 (\\ displaystyle 3), - 2 (\\ displaystyle -2), 14 (\\ displaystyle 14)) into the formula: - b ± b 2 - 4 a c 2 a (\\ displaystyle (\\ frac (-b \\ pm (\\ sqrt (b ^ (2) -4ac))) (2a))) - (- 2) ± ((- 2) 2 - 4 (3) (14) 2 (3) (\\ displaystyle (\\ frac (- (- 2) \\ pm (\\ sqrt (((-2) ^ (2 ) -4 (3) (14)))) (2 (3)))) 2 ± 4 - (12) (14) 6 (\\ displaystyle (\\ frac (2 \\ pm (\\ sqrt (4- (12) (14)))) (6))) 2 ± (4 - 168 6 (\\ displaystyle (\\ frac (2 \\ pm (\\ sqrt ((4-168))) (6))) 2 ± - 164 6 (\\ displaystyle (\\ frac (2 \\ pm (\\ sqrt (-164))) (6)))
• First root: 2 + - 164 6 (\\ displaystyle (\\ frac (2 + (\\ sqrt (-164))) (6))) 2 + 12.8 i 6 (\\ displaystyle (\\ frac (2 + 12.8i) (6)))
• Second root: 2 - 12.8 i 6 (\\ displaystyle (\\ frac (2-12,8i) (6)))

1. Use zero and quadratic roots as solutions to the cubic equation. Quadratic equations have two roots, while cubic ones have three. You have already found two solutions - these are the roots of the quadratic equation. If you put "x" outside the brackets, the third solution is.

   How to find whole roots using multipliers

   1. Make sure there is a free term in the cubic equation d (\\ displaystyle d) . If in an equation of the form a x 3 + b x 2 + c x + d \u003d 0 (\\ displaystyle ax ^ (3) + bx ^ (2) + cx + d \u003d 0) there is a free member d (\\ displaystyle d) (which is not equal to zero), it will not work to put "x" outside the brackets. In this case, use the method outlined in this section.

      Write down the factors of the coefficient a (\\ displaystyle a) and a free member d (\\ displaystyle d) . That is, find the factors of the number at x 3 (\\ displaystyle x ^ (3)) and numbers before the equal sign. Recall that the factors of a number are the numbers that, when multiplied, produce that number.

      Divide each factor a (\\ displaystyle a) for each factor d (\\ displaystyle d) . As a result, you get a lot of fractions and several integers; the roots of the cubic equation will be one of the whole numbers or the negative value of one of the whole numbers.

      • In our example, divide the factors a (\\ displaystyle a) (1 and 2 ) by factors d (\\ displaystyle d) (1 , 2 , 3 and 6 ). You'll get: 1 (\\ displaystyle 1), , , , 2 (\\ displaystyle 2) and. Now add negative values \u200b\u200bof the obtained fractions and numbers to this list: 1 (\\ displaystyle 1), - 1 (\\ displaystyle -1), 1 2 (\\ displaystyle (\\ frac (1) (2))), - 1 2 (\\ displaystyle - (\\ frac (1) (2))), 1 3 (\\ displaystyle (\\ frac (1) (3))), - 1 3 (\\ displaystyle - (\\ frac (1) (3))), 1 6 (\\ displaystyle (\\ frac (1) (6))), - 1 6 (\\ displaystyle - (\\ frac (1) (6))), 2 (\\ displaystyle 2), - 2 (\\ displaystyle -2), 2 3 (\\ displaystyle (\\ frac (2) (3))) and - 2 3 (\\ displaystyle - (\\ frac (2) (3)))... The whole roots of the cubic equation are some numbers from this list.

   2. Plug in integers into the cubic equation. If the equality is true, the substituted number is the root of the equation. For example, substitute in the equation 1 (\\ displaystyle 1):

      Use the method of dividing polynomials by horner's scheme to find the roots of the equation faster. Do this if you don't want to manually substitute numbers into the equation. In Horner's scheme, integers are divided by the values \u200b\u200bof the coefficients of the equation a (\\ displaystyle a), b (\\ displaystyle b), c (\\ displaystyle c) and d (\\ displaystyle d)... If the numbers are evenly divisible (that is, the remainder is), the integer is the root of the equation.

Number e is an important mathematical constant that is the basis of the natural logarithm. Number e approximately equal to 2.71828 with a limit (1 + 1/n)n at n tending to infinity.

Enter the x value to find the exponential function value ex

To calculate numbers with a letter E use an exponential to integer conversion calculator

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Algebra Calculator Calculation

The number e is an important mathematical constant underlying the natural logarithm.

0.3 for power x times 3 for power x are the same

The number e is approximately 2.71828 with a limit of (1 + 1 / n) n for n that goes to infinity.

This number is also called Euler's number or Napier's number.

Exponential - exponential function f (x) \u003d exp (x) \u003d ex, where e is Euler's number.

Enter the value x to find the value of the exponential function ex

Calculating the value of an exponential function in the network.

When Euler's number (e) rises to zero, the answer is 1.

When you raise to a level greater than one, the answer will be greater than the original. If the speed is greater than zero, but less than 1 (for example, 0.5), the answer will be greater than 1, but less than the original (mark E). When the indicator rises to negative power, 1 must be divided by the number e by the given power, but with a plus sign.

Definitions

exhibitor This is an exponential function y (x) \u003d e x, the derivative of which is the same as the function itself.

The indicator is marked with, or.

Number e

The base of the exponent is the number e.

This is an irrational number. It's about the same
e ≈ 2,718281828459045 …

The number e is determined outside the sequence boundary. This is the so-called other exceptional limit:
.

The number e can also be represented as a series:
.

Exhibitor schedule

The graph shows the exponent, e in stage x.
y (x) \u003d ex
The graph shows that it monotonically increases exponentially.

formula

The basic formulas are the same as for an exponential function with a base of level e.

Expression of exponential functions with an arbitrary basis a in the sense of the exponential:
.

see also section "Exponential function" \u003e\u003e\u003e

Private values

Let y (x) \u003d e x.

5 to power x and is equal to 0

Exponential properties

The exponent has the properties of an exponential function with a basis of degree e \u003e first

Definition field, set of values

For x, the exponent y (x) \u003d e x is determined.
Its volume:
— ∞ < x + ∞.
Its meaning:
0 < Y < + ∞.

Extremes, increase, decrease

The exponent is a monotonic increasing function, so it has no extrema.

Its main properties are shown in the table.

Inverse function

The inverse is the natural logarithm.
;
.

Derivative indicators

derivative e in stage x it e in stage x :
.
Derived N-order:
.
Execution of formulas\u003e\u003e\u003e

integral

see also section "Table of Indefinite Integrals" \u003e\u003e\u003e

Complex rooms

Complex number operations are performed using Euler's formula:
,
where is the imaginary unit:
.

Expressions in terms of hyperbolic functions

Expressions in terms of trigonometric functions

Expanding power series

When is x equal to zero?

Regular or online calculator

Regular calculator

The standard calculator gives you simple calculator operations such as addition, subtraction, multiplication and division.

You can use a fast math calculator

Scientific calculator allows you to perform more complex operations as well as calculator such as sine, cosine, inverse sine, inverse cosine, tangent, exponent, exponent, logarithm, interest as well as business in the web memory calculator.

You can enter directly from the keyboard, first click on the area with the calculator.

It performs simple operations on numbers as well as more complex ones like
math calculator online.
0 + 1 = 2.
Here are two calculators:

1. Calculate the first as usual
2. Another calculates it as engineering

The rules are applied to the calculator calculated on the server

Rules for entering terms and functions

Why do I need this online calculator?

Online calculator - how is it different from a regular calculator?

Firstly, a standard calculator is not suitable for transport, and secondly, now the Internet is almost everywhere, this does not mean that there are problems, go to our website and use a web calculator.
Online calculator - how is it different from java calculator as well as other calculators for operating systems?

- again - mobility. If you are on a different computer, you do not need to reinstall it
So, use this site!

Expressions can consist of functions (written in alphabetical order):

absolute (x) Absolute value x
(module x or | x |) arccos (x) Function - arcoxin from xarccosh (x) Arxosin is hyperbolic from xarcsin (x) Separate son xarcsinh (x) HyperX hyperbolic xarctg (x) The function is the arctangent of xarctgh (x) The arctangent is hyperbolic xee number - about 2.7 exp (x) Function - indicator x (as e^x) log (x) or ln (x) Natural logarithm x
(Yes log7 (x), You must enter log (x) / log (7) (or, for example, for log10 (x)\u003d log (x) / log (10)) pi The "Pi" number, which is about 3.14 sin (x) Function - Sine xcos (x) Function - Taper from xsinh (x) Function - Sine hyperbolic xcosh (x) Function - cosine hyperbolic xsqrt (x) The function is the square root of xsqr (x) or x ^ 2 Function - square xtg (x) Function - Tangent of xtgh (x) The function is tangent hyperbolic from xcbrt (x) The function is a cube root xsoil (x) Rounding function x on the underside (soil example (4.5) \u003d\u003d 4.0) symbol (x) Function - symbol xerf (x) Error function (Laplace or Probability Integral)

The following operations can be used in terms of:

Real numbers enter in the form 7,5 , not 7,5 2 * x - multiplication 3 / x - separation x ^ 3 - eksponentiacija x + 7 - Besides, x - 6 - countdown

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Exponential equations are equations of the form

x is an unknown exponent,

a and b- some numbers.

Examples of an exponential equation:

And the equations:

will no longer be indicative.

Consider examples of solving exponential equations:

Example 1.
Find the root of the equation:

Let us bring the degrees to the same base in order to use the property of the degree with a real exponent

Then it will be possible to remove the base of the degree and go to the equality of indicators.

Let's transform the left side of the equation:

We transform the right side of the equation:

We use the property of degree

Answer: 4.5.

Example 2.
Solve the inequality:

Divide both sides of the equation by

Reverse replacement:

Answer: x \u003d 0.

Solve the equation and find the roots at a given interval:

We bring all terms to the same base:

Replacement:

We are looking for the roots of the equation by choosing multiples of the free term:

- suitable because

equality holds.
- suitable because

How to solve? e ^ (x-3) \u003d 0 e to the power of x-3

equality holds.
- suitable because equality holds.
- does not fit, because equality is not satisfied.

Reverse replacement:

A number becomes 1 if its exponent is 0

Doesn't fit because

The right side is 1, because

Hence:

Solve the equation:

Replacement: then

Reverse replacement:

1 equation:

if the bases of the numbers are equal, then their indicators will be equal, then

2 equation:

Logarithm both sides to base 2:

The exponent comes before the expression, because

The left side is 2x because

Hence:

Solve the equation:

Let's transform the left side:

We multiply the degrees by the formula:

Let's simplify: by the formula:

Let's represent in the form:

Replacement:

Let's convert the fraction to an incorrect one:

a2 - not suitable, because

Reverse replacement:

We bring to a common ground:

If a

Answer: x \u003d 20.

Solve the equation:

O.D.Z.

We transform the left side by the formula:

Replacement:

We calculate the root of the discriminant:

a2-not suitable, because

but does not take negative values

We bring to a common ground:

If a

Squaring both sides:

Article editors: Gavrilina Anna Viktorovna, Ageeva Lyubov Alexandrovna

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Translation of a large article "An Intuitive Guide To Exponential Functions & e"

The number e has always worried me - not as a letter, but as a mathematical constant.

What does the number e really mean?

Various math books and even my beloved Wikipedia describe this majestic constant in completely goofy scientific jargon:

The mathematical constant e is the base of the natural logarithm.

If you are interested in what the natural logarithm is, you will find this definition:

The natural logarithm, formerly known as the hyperbolic logarithm, is the base e logarithm, where e is an irrational constant approximately equal to 2.718281828459.

The definitions are, of course, correct.

But it is extremely difficult to understand them. Of course, Wikipedia is not to blame for this: usually mathematical explanations are dry and formal, compiled to the fullest extent of science. Because of this, it is difficult for beginners to master the subject (and once everyone was a beginner).

I'm over it! Today I share my highly intellectual thoughts on what is the number e, and why is it so cool! Put your thick, fearsome math books aside!

The number e is not just a number

Describing e as “a constant of approximately 2.71828 ...” is like calling pi “an irrational number of approximately 3.1415…”.

No doubt it is, but the point still eludes us.

The number pi is the ratio of the circumference to the diameter, the same for all circles... This is a fundamental proportion inherent in all circles, and therefore, it participates in the calculation of the circumference, area, volume and surface area for circles, spheres, cylinders, etc.

Pi shows that all circles are connected, not to mention trigonometric functions derived from circles (sine, cosine, tangent).

The e number is the basic growth ratio for all continuously growing processes. The number e allows you to take a simple growth rate (where the difference is visible only at the end of the year) and calculate the components of this indicator, the normal growth, in which with each nanosecond (or even faster) everything grows a little more.

The number e participates in both exponential and constant growth systems: population, radioactive decay, percentage counting, and many, many others.

Even graded systems that do not grow uniformly can be approximated using the number e.

Also, as any number can be viewed as a "scaled" version 1 (base unit), any circle can be viewed as a "scaled" version of the unit circle (with a radius of 1).

The equation is given: e to the power of x \u003d 0. What is x?

And any growth rate can be viewed as a “scaled” version of e (a “unit” growth rate).

So the number e is not a random number taken at random. The e number embodies the idea that all continuously growing systems are scaled versions of the same metric.

Exponential growth concept

Let's start by looking at a basic system that doubles over a period of time.

For instance:

• Bacteria divide and double in number every 24 hours
• We get twice as many noodles if we break them in half.
• Your money doubles every year if you make 100% profit (lucky!)

And it looks like this:

Dividing or doubling is a very simple progression. Of course, we can triple or quadruple, but doubling is more convenient for clarification.

Mathematically, if we have x divisions, we get 2 ^ x times more good than we had in the beginning.

If only 1 split is done, we get 2 ^ 1 times more. If there are 4 partitions, we get 2 ^ 4 \u003d 16 parts. The general formula looks like this:

In other words, doubling is 100% growth.

We can rewrite this formula like this:

height \u003d (1 + 100%) x

This is the same equality, we only divided "2" into its component parts, which in essence is this number: the initial value (1) plus 100%. Clever, huh?

Of course, we can substitute any other number (50%, 25%, 200%) instead of 100% and get the growth formula for this new coefficient.

The general formula for x periods of a time series will be:

growth \u003d (1 + growth) x

It just means that we use the rate of return, (1 + increment), "x" times in a row.

Let's take a closer look

Our formula assumes that the increment occurs in discrete steps. Our bacteria wait, wait, and then bam !, and at the last minute they double in number. Our profit on interest from the deposit magically appears in exactly 1 year.

Based on the above formula, the profit grows in steps. Green dots appear suddenly.

But the world is not always like that.

If we enlarge the picture, we can see that our bacteria friends are constantly dividing:

The green fellow does not arise out of nothing: it slowly grows out of the blue parent. After 1 period of time (24 hours in our case), the green friend is already fully ripe. Having matured, he becomes a full-fledged blue member of the herd and can create new green cells himself.

Will this information somehow change our equation?

In the case of bacteria, semi-formed green cells still cannot do anything until they grow and separate from their blue parents. So the equation is correct.

In the next article, we'll look at an example of your money growing exponentially.