What equation is called the equation of this line. Cool work04/02/12

Let's review * Which equation is called quadratic? * What equations are called incomplete quadratic equations? * Which quadratic equation is called reduced? * What is called the root of a quadratic equation? * What does it mean to solve a quadratic equation? Which equation is called quadratic? What equations are called incomplete quadratic equations? Which quadratic equation is called reduced? What is the root of a quadratic equation? What does it mean to solve a quadratic equation? Which equation is called quadratic? What equations are called incomplete quadratic equations? Which quadratic equation is called reduced? What is the root of a quadratic equation? What does it mean to solve a quadratic equation?

Algorithm for solving a quadratic equation: 1. Determine the most rational way to solve a quadratic equation 2. Choose the most rational way to solve 3. Determining the number of roots of a quadratic equation 4. Finding the roots of a quadratic equation For better memorization, fill out the table... For better memorization, fill out the table... For better memorization, fill out table...

Additional condition Equation Roots Examples 1. c = c = 0, a 0 ax 2 = 0 x 1 = 0 2. c = 0, a 0, b 0 ax 2 + bx = 0 x 1 = 0, x 2 = -b /a 3. c = 0, a 0, c 0 ax 2 + c = 0 a) x 1.2 = ±(c/a), where c/a 0. b) if c/a 0, then there are no solutions 4. a 0 ax 2 + bx + c = 0 x 1.2 =(-b±D)/2 a, where D = b 2 – 4 ac, D0 5. c – even number (b = 2k), a 0, in 0, c 0 х 2 + 2kx + c = 0 x 1.2 =(-b±D)/а, D 1 = k 2 – ac, where k = 6. The inverse theorem to Vieta’s theorem x 2 + px + q = 0x 1 + x 2 = - p x 1 x 2 = q

II. Special methods 7. Method of isolating the square of a binomial. Goal: Reduce a general equation to an incomplete quadratic equation. Note: the method is applicable to any quadratic equations, but is not always convenient to use. Used to prove the formula for the roots of a quadratic equation. Example: solve the equation x 2 -6 x+8=0 8. Method of “transferring” the highest coefficient. The roots of the quadratic equations ax 2 + bx + c = 0 and y 2 +by+ac=0 are related by the relations: and Note: the method is good for quadratic equations with “convenient” coefficients. In some cases, it allows you to solve a quadratic equation orally. Example: solve the equation 2 x 2 -9 x-5=0 Based on theorems: Example: solve the equation 157 x x-177=0 9. If in a quadratic equation a+b+c=0, then one of the roots is 1, and the second, according to Vieta’s theorem, is equal to c / a 10. If in a quadratic equation a + c = b, then one of the roots is equal to -1, and the second, according to Vieta’s theorem, is equal to -c / a Example: solve the equation 203 x x + 17 = 0 x 1 =y 1 /a, x 2 =y 2 /a

III. General methods for solving equations 11. Factorization method. Goal: Reduce a general quadratic equation to the form A(x)·B(x)=0, where A(x) and B(x) are polynomials with respect to x. Methods: Taking the common factor out of brackets; Using abbreviated multiplication formulas; Grouping method. Example: solve the equation 3 x 2 +2 x-1=0 12. Method of introducing a new variable. Good choice of a new variable makes the structure of the equation more transparent Example: solve the equation (x 2 +3 x-25) 2 -6(x 2 +3 x-25) = - 8

Equality of the form F (x, y) = 0 called an equation in two variables x, y, if it is not true for all pairs of numbers x, y. They say two numbers x = x 0 , y=y 0, satisfy some equation of the form F(x, y)=0, if when substituting these numbers instead of variables X And at in the equation, its left side vanishes.

The equation of a given line (in a designated coordinate system) is an equation with two variables that is satisfied by the coordinates of every point lying on this line and not satisfied by the coordinates of every point not lying on it.

In what follows, instead of the expression “the equation of the line is given F(x, y) = 0" we will often say in short: given a line F (x, y) = 0.

If the equations of two lines are given F(x, y) = 0 And Ф(x, y) = Q, then the joint solution of the system

gives all their intersection points. More precisely, each pair of numbers that is a joint solution of this system determines one of the intersection points.

*) In cases where the coordinate system is not named, it is assumed that it is Cartesian rectangular.

157. Points are given *) M 1 (2; - 2), M 2 (2; 2), M 3 (2; - 1), M 4 (3; -3), M 5 (5; -5), M 6 (3; -2). Determine which published points lie on the line defined by the equation X+ y = 0, and which ones do not lie on it. Which line is defined by this equation? (Draw it on the drawing.)

158. On the line defined by the equation X 2 +y 2 =25, find the points whose abscissas are equal to the following numbers: a) 0, b) - 3, c) 5, d) 7; on the same line find points whose ordinates are equal to the following numbers: e) 3, f) - 5, g) - 8. Which line is determined by this equation? (Draw it on the drawing.)

159. Determine which lines are determined by the following equations (construct them on the drawing):

1) x - y = 0; 2) x + y = 0; 3) x- 2 = 0; 4) x+ 3 = 0;

5) y - 5 = 0; 6) y+ 2 = 0; 7) x = 0; 8) y = 0;

9) x 2 - xy = 0; 10) xy+ y 2 = 0; eleven) x 2 - y 2 = 0; 12) xy= 0;

13) y 2 - 9 = 0; 14) xy 2 - 8xy+15 = 0; 15) y 2 +5y+4 = 0;

16) X 2 y - 7xy + 10y = 0; 17) y =|x|; 18) x =|at|; 19)y + |x|=0;

20) x +|at|= 0; 21)y =|X- 1|; 22) y = |x+ 2|; 23) X 2 + at 2 = 16;

24) (x-2) 2 +(y-1) 2 =16; 25) (x+ 5) 2 +(y- 1) 2 = 9;

26) (X - 1) 2 + y 2 = 4; 27) x 2 +(y + 3) 2 = 1; 28) (x -3) 2 + y 2 = 0;

29) X 2 + 2y 2 = 0; 30) 2X 2 + 3y 2 + 5 = 0

31) (x- 2) 2 + (y + 3) 2 + 1=0.

160.Lines given:

1)X+ y = 0; 2)x - y = 0; 3) x 2 + y 2 - 36 = 0;

4) x 2 +y 2 -2x==0; 5) x 2 +y 2 + 4x-6y-1 =0.

Determine which of them pass through the origin.

161.Lines given:

1) x 2 + y 2 = 49; 2) (x- 3) 2 + (y+ 4) 2 = 25;

3) (x+ 6) 2 + (y - 3) 2 = 25; 4) ( x + 5) 2 + (y - 4) 2 = 9;

5) x 2 +y 2 - 12x + 16y = 0; 6) x 2 +y 2 - 2x + 8at+ 7 = 0;

7) x 2 +y 2 - 6x + 4y + 12 = 0.

Find their points of intersection: a) with the axis Oh; b) with an axis OU.

162.Find the intersection points of two lines;

1)X 2 +y 2 = 8, x-y = 0;

2) X 2 +y 2 -16x+4at+18 = 0, x + y= 0;

3) X 2 +y 2 -2x+4at -3 = 0, X 2 + y 2 = 25;

4) X 2 +y 2 -8x+10у+40 = 0, X 2 + y 2 = 4.

163. Points are given in the polar coordinate system

M 1 (1; ), M 2 (2; 0), M 3 (2; )

M 4 (
;) And M 5 (1; )

Determine which of these points lie on the line defined by the equation in polar coordinates  = 2 cos , and which do not lie on it. Which line is determined by this equation? (Draw it on the drawing:)

164. On the line defined by the equation  = , find points whose polar angles are equal to the following numbers: a) ,b) - , c) 0, d) . Which line is defined by this equation?

(Build it on the drawing.)

165.On the line defined by the equation  = , find points whose polar radii are equal to the following numbers: a) 1, b) 2, c)
. Which line is defined by this equation? (Build it on the drawing.)

166. Establish which lines are determined in polar coordinates by the following equations (construct them on the drawing):

1)  = 5; 2)  = ; 3)  = ; 4)  cos  = 2; 5)  sin  = 1;

6)  = 6 cos ; 7)  = 10 sin ; 8) sin  = 9) sin  =

167. Construct the following Archimedes spirals on the drawing:

1)  = 5, 2)  = 5; 3)  = ; 4)р = -1.

168. Construct the following hyperbolic spirals on the drawing:

1)  = ; 2) = ; 3) = ; 4) = - .

169. Construct the following logarithmic spirals on the drawing:

,
.

170. Determine the lengths of the segments into which the Archimedes spiral cuts

ray emerging from the pole and inclined to the polar axis at an angle
. Make a drawing.

171. On the Archimedes spiral
point taken WITH, whose polar radius is 47. Determine how many parts this spiral cuts the polar radius of the point WITH, Make a drawing.

172. On a hyperbolic spiral
find a point R, whose polar radius is 12. Make a drawing.

173. On a logarithmic spiral
find point Q whose polar radius is 81. Make a drawing.

A line on a plane is a collection of points on this plane that have certain properties, while points that do not lie on a given line do not have these properties. The equation of a line defines an analytically expressed relationship between the coordinates of points lying on this line. Let this relationship be given by the equation

F( x,y)=0. (2.1)

A pair of numbers satisfying (2.1) is not arbitrary: if X given, then at cannot be anything, meaning at associated with X. When it changes X changes at, and a point with coordinates ( x,y) describes this line. If the coordinates of point M 0 ( X 0 ,at 0) satisfy equation (2.1), i.e. F( X 0 ,at 0)=0 is a true equality, then point M 0 lies on this line. The converse is also true.

Definition. An equation of a line on a plane is an equation that is satisfied by the coordinates of any point lying on this line, and not satisfied by the coordinates of points not lying on this line.

If the equation of a certain line is known, then the study of the geometric properties of this line can be reduced to the study of its equation - this is one of the main ideas of analytical geometry. To study equations, there are well-developed methods of mathematical analysis that simplify the study of the properties of lines.

When considering lines the term is used current point line – variable point M( x,y), moving along this line. Coordinates X And at current point are called current coordinates line points.

If from equation (2.1) we can express explicitly at
through X, that is, write equation (2.1) in the form , then the curve defined by such an equation is called schedule functions f(x).

1. The equation is given: , or . If X takes arbitrary values, then at takes values ​​equal to X. Consequently, the line defined by this equation consists of points equidistant from the coordinate axes Ox and Oy - this is the bisector of the I–III coordinate angles (straight line in Fig. 2.1).

The equation, or, determines the bisector of the II–IV coordinate angles (straight line in Fig. 2.1).

0 x 0 x C 0 x

rice. 2.1 fig. 2.2 fig. 2.3

2. The equation is given: , where C is some constant. This equation can be written differently: . This equation is satisfied by those and only those points, ordinates at which are equal to C for any abscissa value X. These points lie on a straight line parallel to the Ox axis (Fig. 2.2). Similarly, the equation defines a straight line parallel to the Oy axis (Fig. 2.3).

Not every equation of the form F( x,y)=0 defines a line on the plane: the equation is satisfied by a single point – O(0,0), and the equation is not satisfied by any point on the plane.

In the examples given, we used a given equation to construct a line determined by this equation. Let's consider the inverse problem: construct its equation using a given line.

3. Create an equation for a circle with center at point P( a,b) And
radius R .

○ A circle with a center at point P and radius R is a set of points located at a distance R from point P. This means that for any point M lying on the circle, MP = R, but if point M does not lie on the circle, then MP ≠ R.. ●

Straight line on a plane and in space.

The study of the properties of geometric figures using algebra is called analytical geometry , and we will use the so-called coordinate method .

A line on a plane is usually defined as a set of points that have properties unique to them. The fact that the x and y coordinates (numbers) of a point lying on this line are written analytically in the form of some equation.

Def.1 Equation of a line (equation of a curve) on the Oxy plane is called an equation (*), which is satisfied by the x and y coordinates of each point on a given line and is not satisfied by the coordinates of any other point not lying on this line.

From Definition 1 it follows that every line on the plane corresponds to some equation between the current coordinates ( x,y ) points of this line and vice versa, each equation corresponds, generally speaking, to a certain line.

This gives rise to two main problems of analytical geometry on the plane.

1. A line is given in the form of a set of points. We need to create an equation for this line.

2. The equation of the line is given. It is necessary to study its geometric properties (shape and location).

Example. Do the points lie A(-2;1) And IN (1;1) on line 2 X +at +3=0?

The problem of finding the intersection points of two lines given by the equations and comes down to finding coordinates that satisfy the equation of both lines, i.e. to solving a system of two equations with two unknowns.

If this system has no real solutions, then the lines do not intersect.

The concept of a line is introduced in the UCS in a similar way.

A line on a plane can be defined by two equations

Where X And at – arbitrary point coordinates M(x;y), lying on this line, and t - a variable called parameter , the parameter determines the position of the point on the plane.

For example, if , then the value of the parameter t=2 corresponds to the point (3;4) on the plane.

If the parameter changes, the point on the plane moves, describing this line. This method of defining a line is called parametric, and equation (5.1) is a parametric equation of the line.

To move from parametric equations to a general equation (*), one must somehow eliminate the parameter from the two equations. However, we note that such a transition is not always advisable and not always possible.

A line on a plane can be specified vector equation , where t is a scalar variable parameter. Each parameter value corresponds to a specific plane vector. When changing the parameter, the end of the vector will describe a certain line.

Vector equation in DSC corresponds two scalar equations

(5.1), i.e. the equation of projections on the coordinate axes of the vector equation of a line is its

parametric equation.

The vector equation and the parametric line equations have a mechanical meaning. If a point moves on a plane, then the indicated equations are called equations of motion , and the line is the trajectory of the point, the parameter t is time.

Conclusion: every line on the plane corresponds to an equation of the form.

In the general case, ANY EQUATION OF A VIEW corresponds to a certain line, the properties of which are determined by the given equation (with the exception that no geometric image corresponds to an equation on a plane).

Let a coordinate system on the plane be chosen.

Def. 5.1. Line equation This type of equation is calledF(x;y) =0, which is satisfied by the coordinates of every point lying on this line, and not satisfied by the coordinates of any point not lying on it.

Equation of the formF(x;y )=0 – called the general equation of a line or an equation in implicit form.

Thus, line Г is the locus of points satisfying this equation Г=((x, y): F(x;y)=0).

The line is also called crooked.

Solving the equation

Illustration of a graphical method for finding the roots of an equation

Solving an equation is the task of finding such values ​​of the arguments at which this equality is achieved. Additional conditions (integer, real, etc.) can be imposed on the possible values ​​of the arguments.

Substituting another root produces an incorrect statement:

.

Thus, the second root must be discarded as extraneous.

Types of equations

There are algebraic, parametric, transcendental, functional, differential and other types of equations.

Some classes of equations have analytical solutions, which are convenient because they not only give the exact value of the root, but also allow you to write the solution in the form of a formula, which can include parameters. Analytical expressions allow not only to calculate the roots, but also to analyze their existence and their quantity depending on the parameter values, which is often even more important for practical use than the specific values ​​of the roots.

Equations for which analytical solutions are known include algebraic equations of no higher than the fourth degree: linear equation, quadratic equation, cubic equation and fourth degree equation. Algebraic equations of higher degrees in the general case do not have an analytical solution, although some of them can be reduced to equations of lower degrees.

An equation that includes transcendental functions is called transcendental. Among them, analytical solutions are known for some trigonometric equations, since the zeros of trigonometric functions are well known.

In the general case, when an analytical solution cannot be found, numerical methods are used. Numerical methods do not provide an exact solution, but only allow one to narrow the interval in which the root lies to a certain predetermined value.

Examples of equations

see also

Literature

• Bekarevich, A. B. Equations in a school mathematics course / A. B. Bekarevich. - M., 1968.
• Markushevich, L. A. Equations and inequalities in the final repetition of the high school algebra course / L. A. Markushevich, R. S. Cherkasov. / Mathematics at school. - 2004. - No. 1.
• Kaplan Y. V. Rivnyannya. - Kyiv: Radyanska School, 1968.
• The equation- article from the Great Soviet Encyclopedia
• Equations// Collier's Encyclopedia. - Open society. 2000.
• The equation// Encyclopedia Around the World
• The equation// Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

Links

• EqWorld - World of Mathematical Equations - contains extensive information about mathematical equations and systems of equations.

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