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How to solve cubic equations. Quadratic inequalities Expressions using trigonometric functions

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

1. x 2 -8x+12 ≥ 0

2. -x 2 +3x > 0

3. x 2 ≤ 4

Well, you understand...)

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

The inequality ready for solution has the form: on the left is a quadratic trinomial ax 2 +bx+c, on the right - zero. The inequality sign can be absolutely anything. The first two examples are here are already ready to make a decision. The third example still needs to be prepared.

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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. Let's learn - 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 has the form . Some cubic equations are not that easy to solve, but if you use the right method (with good theoretical background), you can find the roots of even the most complex cubic equation - to do this, use the formula for solving a quadratic equation, find whole roots, or calculate the discriminant.

Steps

How to solve a cubic equation without a free term

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

    Bracket out x (\displaystyle x) . Since there is no free term in the equation, each term of the equation includes a variable x (\displaystyle x). This means that one x (\displaystyle x) can be taken out of brackets 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 = 0 (\displaystyle ax^(2)+bx+c=0) can be factorized. This equation will be obtained if we take out x (\displaystyle x) out of brackets. In our example:

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

    • In our example, substitute the values ​​of 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 the zero and roots of a quadratic equation as solutions to a cubic equation. Quadratic equations have two roots, while cubic equations have three. You have already found two solutions - these are the roots of the quadratic equation. If you took “x” out of brackets, the third solution would be .

    How to find whole roots using factors

    1. Make sure there is an intercept in the cubic equation d (\displaystyle d) . If in an equation of the form a x 3 + b x 2 + c x + d = 0 (\displaystyle ax^(3)+bx^(2)+cx+d=0) have a free member d (\displaystyle d)(which is not zero), putting “x” out of brackets will not work. IN in this case use the method outlined in this section.

      Write down the coefficient factors a (\displaystyle a) and free member d (\displaystyle d) . That is, find the factors of the number when x 3 (\displaystyle x^(3)) and numbers before the equals 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 multiplier d (\displaystyle d) . The end result is a lot of fractions and a few whole numbers; The roots of a cubic equation will be one of the integers or the negative value of one of the integers.

      • 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 to this list negative values resulting fractions and numbers: 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 integer roots of a cubic equation are some numbers from this list.

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

      Use the method of dividing polynomials by Horner's scheme to quickly find the roots of the equation. Do this if you don't want to manually plug in numbers into the equation. In Horner's scheme, integers are divided by the values ​​of the coefficients of the equation a (\displaystyle a), b (\displaystyle b), c (\displaystyle c) And d (\displaystyle d). If the numbers are divisible by an integer (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 limit (1 + 1/n)n at n tending to infinity.

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

To calculate numbers with a letter E use 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 at power x times 3 at 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) = exp (x) = ex, where e is Euler's number.

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

Calculating the value of an exponential function in a network.

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

When you raise to more than one level, 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 increases to negative power, 1 must be divided by the number e per given power, but with a plus sign.

Definitions

exhibitor This is an exponential function y (x) = e x, the derivative of which coincides with the function itself.

The indicator is marked as, 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 beyond the boundary of the sequence. This is the so-called other exceptional limit:
.

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

Exponential graph

The graph shows the exponent, e in progress X.
y(x) = ex
The graph shows that it increases monotonically exponentially.

formula

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

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

also department "Exponential function" >>>

Private values

Let y(x) = e x.

5 to power x and equals 0

Exponential properties

The indicator has the properties of an exponential function with a basis of degree e> first

Definition field, value set

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

Extremes, increase, decrease

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

Its main properties are shown in the table.

Inverse function

The reciprocal is the natural logarithm.
;
.

Derivatives of indicators

derivative e in progress X This e in progress X :
.
Derived N-order:
.
Executing formulas > > >

integral

also section "Table of indefinite integrals" >>>

Complex numbers

Operations with complex numbers are performed using Euler's formula:
,
where is the imaginary unit:
.

Expressions through hyperbolic functions

Expressions using trigonometric functions

Expansion of power series

When is x equal to zero?

Regular or online calculator

Regular calculator

Standard Calculator gives you simple calculator operations like adding, subtracting, multiplying and dividing.

You can use a quick math calculator

Scientific calculator allows you to perform more complex operations as well as calculator like sine, cosine, inverse sine, inverse cosine which is tangent, tangent, exponent, exponent, logarithm, interest and also business in web memory calculator.

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

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

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

The rules apply to the calculator calculated on the server

Rules for entering terms and functions

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

Firstly, the 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 the web calculator.
Online calculator - how it differs from a java calculator, as well as from 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 (noted in alphabetical order):

absolute(x) Absolute value X
(module X or | x |) arccos(x) Function - arcoxin from Xarccosh(x) Arxosine is a hyperbolic of Xarcsin(x) Separate son Xarcsinh(x) HyperX hyperbolic Xarctan(x) Function is the arctangent of Xarctgh(x) The arctangent is hyperbolic Xee number - about 2.7 exp(x) Function - indicator X(How e^X) log(x) or ln(x) Natural logarithm X
(Yes log7(x) You must enter log(x)/log(7) (or for example, log10(x)= log(x)/log(10)) pi The number "Pi", which is about 3.14 sin(x) Function - Sine Xcos(x) Function - Cone from Xsinh(x) Function - Hyperbolic sine Xcosh(x) Function - cosine-hyperbolic Xsqrt(x) The function is Square root from Xsqr(x) or x^2 Function - square Xtg(x) Function - Tangent from Xtgh(x) The function is a hyperbolic tangent from Xcbrt(x) The function is the cube root Xsoil (x) Rounding function X on the bottom side (soil example (4.5) == 4.0) character (x) Function - symbol Xerf(x) Error function (Laplace or probability integral)

The following operations can be used in terms:

Real numbers enter in the form 7,5 , Not 7,5 2*x- multiplication 3/x- division 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 exponential equation:

And the equations:

will no longer be indicative.

Let's look at examples of solving exponential equations:

Example 1.
Find the root of the equation:

Let's reduce the powers to the same base to take advantage of the property of powers with a real exponent

Then it will be possible to remove the base of the degree and move on to equality of exponents.

Let's transform the left side of the equation:


Let's transform the right side of the equation:

Using the property of degree

Answer: 4.5.

Example 2.
Solve the inequality:

Let's divide both sides of the equation by

Reverse replacement:

Answer: x=0.

Solve the equation and find the roots on the given interval:

We reduce all terms to the same base:

Replacement:

We look for the roots of the equation by selecting multiples of the free term:

– suitable, because

equality is satisfied.
– suitable, because

How to solve? e^(x-3) = 0 e to the power x-3

equality is satisfied.
– suitable, because equality is satisfied.
– not suitable, because equality is not satisfied.

Reverse replacement:

A number becomes 1 if its exponent is 0

Not suitable because

The right side is equal to 1, because

From here:

Solve the equation:

Replacement: , then

Reverse replacement:

1 equation:

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

2 equation:

Let's logarithm both sides to base 2:

The exponent comes before the expression, because

The left side is 2x, because

From here:

Solve the equation:

Let's transform the left side:

We multiply the degrees using the formula:

Let's simplify: according to the formula:

Let's present it in the form:

Replacement:

Let's convert the fraction to improper:

a2 - not suitable, because

Reverse replacement:

Let's get to the general point:

If

Answer: x=20.

Solve the equation:

O.D.Z.

Let's transform the left side using the formula:

Replacement:

We calculate the root of the discriminant:

a2-not suitable, because

but does not take negative values

Let's get to the general point:

If

We square both sides:

Editors of the article: Gavrilina Anna Viktorovna, Ageeva Lyubov Aleksandrovna

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Translation of the large article “An Intuitive Guide To Exponential Functions & e”

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

What does the number e really mean?

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

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

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

The natural logarithm, formerly known as the hyperbolic logarithm, is a logarithm with base e, 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 according to the full rigor of science. This makes it difficult for beginners to master the subject (and everyone was a beginner at one point).

I'm over it! Today I am sharing my highly intelligent thoughts on... what is the number e, and why it’s so cool! Put your thick, intimidating math books aside!

The number e is not just a number

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

This is undoubtedly true, but the point still eludes us.

Pi is the ratio of the circumference to the diameter, the same for all circles. It is a fundamental proportion common to all circles and hence is involved in calculating circumference, area, volume and surface area for circles, spheres, cylinders, etc.

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

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

The number e is involved in both exponential and constant growth systems: population, radioactive decay, percentage calculation, and many, many others.

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

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

The equation is given: e to the power x = 0. What is x equal to?

And any growth factor can be viewed as a "scaled" version of e (the "unit" growth factor).

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

Concept of exponential growth

Let's start by reviewing basic system, which doubles over a certain period of time.

For example:

  • 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 something like this:

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

Mathematically, if we have x divisions, we end up with 2^x times more good than we started with.

If only 1 partition is made, we get 2^1 times more. If there are 4 partitions, we get 2^4=16 parts. The general formula looks like this:

In other words, a doubling is a 100% increase.

We can rewrite this formula like this:

height = (1+100%)x

This is the same equality, we just divided “2” into its component parts, which in essence is this number: the initial value (1) plus 100%. Smart, right?

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 the time series will be:

growth = (1+growth)x

This simply means that we use the return rate, (1 + gain), "x" times in a row.

Let's take a closer look

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

Based on the formula written above, profits grow in steps. Green dots appear suddenly.

But the world is not always like that.

If we zoom in, we can see that our bacterial friends are constantly dividing:

The green fellow does not arise out of nothing: he 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 change our equation in any way?

In the case of bacteria, half-formed green cells still can't do anything until they grow up and separate completely from their blue parents. So the equation is correct.

In the next article we will look at an example of exponential growth of your money.