In a cubic equation, the highest exponent is 3, such an equation has 3 roots (solutions) and it looks like . Some cubic equations are not so easy to solve, but if you apply the right method (with good theoretical preparation), you can find the roots of even the most complex cubic equation - to do this, use the formula for solving a quadratic equation, find integer roots or calculate the discriminant.
Steps
How to solve a cubic equation without a free term
- 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)))
-
Use zero and the roots of the quadratic equation as solutions to the 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 put "x" out of brackets, the third solution is .
How to find integer roots using multipliers
-
Make sure the cubic equation has an intercept 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 equal to zero), it will not work to put "x" out of brackets. In this case, use the method described in this section.
Write out coefficient multipliers a (\displaystyle a) and free member d (\displaystyle d) . That is, find the factors of the number when x 3 (\displaystyle x^(3)) and the numbers before the equals sign. Recall that the factors of a number are the numbers that, when multiplied, give that number.
Divide each multiplier a (\displaystyle a) for every multiplier d (\displaystyle d) . The result will be many fractions and several integers; 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 the negative values of the resulting 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 integer roots of the cubic equation are some numbers from this list.
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Plug in the integers into the cubic equation. If this equality is observed, the substituted number is the root of the equation. For example, plug into the equation 1 (\displaystyle 1):
Use the method of dividing polynomials by Horner's scheme to quickly find the roots of an equation. Do this if you don't want to manually plug 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 evenly divisible (that is, the remainder is ), the integer is the root of the equation.
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Find out if there is an intercept in a cubic equation 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 only the term x 3 (\displaystyle x^(3))(that is, there may be no other members at all).
Take it out of brackets x (\displaystyle x) . Since there is no free term in the equation, each term in the equation includes a variable x (\displaystyle x). This means that one x (\displaystyle x) can be parenthesized to simplify the equation. Thus, the equation will be written as follows: x (a x 2 + b x + c) (\displaystyle x(ax^(2)+bx+c)).
Factorize (by 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. Such an equation will be obtained if x (\displaystyle x) for 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) plug into the formula.
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 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 that underlies the natural logarithm.
0.3 at power x multiplied by 3 by power x are the same
The number e is approximately 2.71828 with a limit of (1 + 1/n)n for n going to infinity.
This number is also called the Euler number or the Napier number.
Exponential - An exponential function f (x) = exp (x) = ex, where e is the Euler number.
Enter the value of x to find the value of the exponential function ex
Calculation of the value of the exponential function in the network.
When the Euler number (e) goes up 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 (e.g. 0.5), the answer will be greater than 1 but less than the original (mark E). When the exponent increases to a negative power, 1 must be divided by the number e for a given power, but with a plus sign.
Definitions
exhibitor This is an exponential function y (x) = e x, whose derivative is the same as the function itself.
The indicator is marked as, or.
e number
The base of the exponent is e.
This is an irrational number. It's about the same
e ≈ 2,718281828459045 …
The number e is defined outside the sequence boundary. This is the so-called other exceptional limit:
.
The number e can also be represented as a series:
.
Exhibitor chart
The graph shows the degree e in stage X.
y(x) = ex
The graph shows that it monotonically increases exponentially.
formula
The basic formulas are the same as for the exponential function with the base level e.
Expression of exponential functions with an arbitrary basis a in the sense of the exponent:
.
also section "Exponential function" >>>
private values
Let y (x) = e x.
5 to power x and equals 0
Exponential Properties
The exponent has the properties of an exponential function with a degree basis e> first
Definition field, set of values
For x, the index y (x) = 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 extremes.
Its main properties are shown in the table.
Inverse function
The reciprocal is the natural logarithm.
;
.
Derivatives of indicators
derivative e in stage X it e in stage X
:
.
Derived N-order:
.
Execution of formulas > > >
integral
also section "Table of indefinite integrals" >>>
Complex rooms
Operations with complex numbers are performed using Euler formula:
,
where is the imaginary unit:
.
Expressions in terms of hyperbolic functions
Expressions in terms of trigonometric functions
Power Series Extension
When is x equal to zero?
Regular or online calculator
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It performs simple operations on numbers as well as more complex ones such as
math calculator online.
0 + 1 = 2.
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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 is it different from java calculator and also from other calculators for operating systems?
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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) Arcosine is hyperbolic Xarcsin(x) Separate son Xarcsinh(x) HyperX hyperbolic Xarctg(x) The function is the arc tangent of Xarctgh(x) 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), Need to type log(x) / log(7) (or e.g. for 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 - Sine hyperbolic Xcash(x) Function - cosine-hyperbolic Xsquare(x) The function is the square root of Xsqr(x) or x^2 Function - square Xtg(x) Function - Tangent from Xtgh(x) The function is a hyperbolic tangent of Xcbrt(x) The function is a cube root Xsoil (x) Rounding function X on the underside (soil example (4.5) == 4.0) symbol(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- separation x^3— exponentiacija x + 7- Besides, x - 6- countdown
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Exponential equations are equations of the form
x - unknown exponent,
a and b- some numbers.
Examples of exponential equation:
And the equations:
will no longer be representative.
Consider examples of solving exponential equations:
Example 1
Find the root of the equation:
We reduce 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 proceed to the equality of indicators.
Let's transform the left side of the equation:
Let's transform the right side of the equation:
Using the degree property
Answer: 4.5.
Example 2
Solve the inequality:
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 are looking for the roots of the equation by selecting multiples of the free term:
- suitable, because
equality holds.
- suitable, because
How to decide? e^(x-3) = 0 e to the power of x-3
equality holds.
- suitable, because equality holds.
- not suitable, because equality is not met.
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:
Logarithm of both parts 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 according to the formula:
Let's simplify: according to the formula:
Let's put it in the form:
Replacement:
Let's convert the fraction to an improper one:
a2 - not suitable, because
Reverse replacement:
Let's get to the bottom line:
If a
Answer: x=20.
Solve the equation:
O.D.Z.
Let's transform the left side according to the formula:
Replacement:
We calculate the root of the discriminant:
a2-does not fit, because
does not take negative values
Let's get to the bottom line:
If a
Let's square both sides:
Article editors: Gavrilina Anna Viktorovna, Ageeva Lyubov Alexandrovna
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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 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 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 about what is e number and why is it 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...".
No doubt it is, but the essence still eludes us.
The number pi is the ratio of the circumference of a circle to its diameter, the same for all circles.. This is a fundamental proportion common to all circles, and therefore, it is involved in calculating the circumference, area, volume and surface area for circles, spheres, cylinders, etc.
Pi shows that all circles are connected, 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 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, normal growth, in which every nanosecond (or even faster) everything grows by a little more.
The number e is involved in both exponential and constant growth systems: population, radioactive decay, interest calculation, and many, many others.
Even stepped systems that do not grow uniformly can be approximated by 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 (radius 1).
An equation is given: e to the power of x \u003d 0. What is x equal to?
And any growth factor can be considered as a "scaled" version of e (a "single" growth factor).
So the number e is not a random number taken at random. The number e embodies the idea that all continuously growing systems are scaled versions of the same metric.
The concept of exponential growth
Let's start by looking at a basic system that doubles over a given period of time.
For example:
- Bacteria divide and "doubling" in numbers every 24 hours
- We get twice as many noodles if we break them in half
- Your money doubles every year if you get 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 get 2^x times more good than we had at the beginning.
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:
growth = (1+100%)x
This is the same equality, we just divided "2" into its component parts, which in essence this number is: 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 ratio.
The general formula for x periods of the time series will look like:
growth = (1+growth)x
This simply means that we use the rate of return, (1 + growth), "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 from 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 this.
If we zoom in, we can see that our bacteria friends are constantly dividing:
The green kid doesn't come 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, the half-formed green cells still can't do anything until they grow up and completely separate from their blue parents. So the equation is correct.
In the next article, we will look at an example of the exponential growth of your money.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
What "square inequality"? Not a question!) If you take any quadratic equation and change the sign in it "=" (equal) to any inequality icon ( > ≥ < ≤ ≠ ), we get a quadratic inequality. For example:
1. x2 -8x+12 ≥ 0
2. -x 2 +3x > 0
3. x2 ≤ 4
Well, you get the idea...)
I knowingly linked equations and inequalities here. The fact 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, look at 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 anything. The first two examples are here are ready for 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.)
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you can get acquainted with functions and derivatives.
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