Incomplete quadratic equations

Presentation of an algebra lesson in grade 8 on the topic "Quadric equations. Solving incomplete quadratic equations." Introduction of the concept of complete and incomplete quadratic equations. Primary consolidation of methods for solving incomplete quadratic equations.

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Presentation of an algebra lesson in grade 8 “Quadric equations. Solving incomplete quadratic equations »

Mysterious, but familiar to us, There is something unknown in it Its root - this is what we are looking for Finding it is interesting to everyone Everyone will say without a doubt Before you (equation)

Solve equations a) y - 7 = 0; b) x + 0.5 = 0; c) a x = 0; d) 2 x - 1/3 = 0; e) a (a - 1) = 0; e) x 2 + 4 = 0.

Task In the cinema hall, the number of seats in each row is 8 more than the number of rows. In total, 884 spectators came to the session and all the seats were taken. How many rows are there in the cinema?

x - rows; x +8 - places in each row C leave the equation: x (x + 8) \u003d 884; x 2 +8x-884=0.

" Quadratic equations. Solving incomplete quadratic equations »Lesson topic: epigraph: an equation is a key that can open a thousand doors to the unknown.

purpose: to introduce the concept of a quadratic equation; Learn how to solve incomplete quadratic equations.

Definition of a quadratic equation A quadratic equation is an equation of the form ax²+bx+c=0 , where x is a variable, a, b, c are parameters, a≠0. The number a is called the first coefficient, the number b is called the second coefficient, and c is the free member. A quadratic equation is also called an equation of the second degree, since its head part is a polynomial of the second degree.

Examples of quadratic equations: a b c -2x²+x-1.4=0 -2 1 -1.4 5x²-4x=0 5 -4 0 3X²+10.3=0 3 0 10.3

Task 1 Are these equations quadratic? 4x²-5x+2=0 -5.6x²-2x- 0.5 =0 13-7x²=0 16x²-x³-5=0 1-16x=0 -x²=0

Task 2 Name the coefficients in the quadratic equation. 3x²-6x+2=0 -x²+5x+10=0 x²-8x+1.5=0 -4x²+5=0 -36x²-3x=0 12x²=0

Incomplete quadratic equations If in a quadratic equation ax² + bx + c \u003d 0 at least one of the coefficients in b or c is equal to zero, then such an equation is called incomplete m square equation. a b c -3x²+5=0 -3 0 5 2x²-10x=0 2 -10 0 16x²=0 16 0 0

Classification of quadratic equations complete incomplete Al-Khwarizmi, where a ≠ 0 b=0 b=0, c=0 c=0 or or or

Solve the equation if b=0. -4x²+25=0 - 4x² =- 25 4x² = 25 or I

Solve the equation if b=0 ,c=0. III

Solve the equation if C=0 . (35 + y) y = 0 35 + y = 0 or II y = 0 y=-35

Testing

one . 2. 3. 4. 5 0; -5 -5; 5 0 Task №1. Give the roots of the equation help

For task number 2. Indicate the roots of the equation 1. 2. 3. 4. -4; 4 - 4; 0 16 0 ; 4

For task number 3. Specify the roots of equation 1 . 2. 3. 4. 3 -3; 0 -3 0 ; 3

For task number 4. Indicate the roots of the equation 1. 2. 3. 4. 0; 4 16 -4; 4-4; 0

01/05/17 Assignment No. 5. Indicate the roots of the equation 1. 2. 3. 4. -2; 2 4 2 2; 0

Lesson summary: Today at the lesson I learned ... I understood ... I learned ... my successes are ... I felt difficulties ... I didn’t know how, but now I can ... at the next lesson I want to ...


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Lesson on the topic "Incomplete quadratic equations". Prepared by the mathematics teacher of the MOU "Uspenskaya secondary school MO" Akhtubinsky district" Zenina N.G., Kramarenko T.N.

"I have to divide my time between politics and equations. However, I think equations are much more important, because politics exists only for this moment, and equations will exist forever." A. Einstein.

Hello guys! Let's repeat: I am your assistant, I will guide you through the whole big topic of Quadratic Equations. In 7th and 8th grade, you already considered and even solved quadratic equations.

Today you will learn: 1. What equations are called quadratic? 2. What is the main thing in the definition of a quadratic equation that should be remembered and taken into account? 3. What special cases of quadratic equations are there? 4. What are the ways to solve quadratic equations in each particular case? Now let's look for answers to these questions together. Good luck!

What do these equations have in common?

A quadratic equation is an equation of the form ... ax ² + bx + c \u003d 0, where a ≠ 0, x is a variable, a, b, c are some numbers. a is the senior (first) coefficient, b is the second coefficient, c is a free term. a is the senior (first) coefficient, b is the second coefficient, c is a free term. a is the senior (first) coefficient, c is the second coefficient, c is the free term.

If a \u003d 1, then the quadratic equation x ² + bx + c \u003d 0 is called reduced. We will solve No. 513 (orally).

and in c 5x² + 5x - 3 \u003d 0 3 x² + 2 x - 4 \u003d 0 x² + 4x + 3 \u003d 0 -2 x² + x - 1 \u003d 0 4 x ² - 4 x + 1 \u003d 0 5 5 -3 3 2 -4 1 4 3 -2 1 - 1 4 - 4 1 Let's try to solve:

I wonder what will happen if the coefficients of the quadratic equation in turn or all at once (except a) turn into zeros. Let's do some research.

Incomplete quadratic equations

Consider all possible cases

Incomplete quadratic equations of the form: no roots.

Incomplete quadratic equations of the form:

Answer: x \u003d 0. there are no roots. Write incomplete quadratic equations:

Write down the quadratic equations with the indicated coefficients: a=1, b=0, c=16; a=-1, b=5, c=0; b=0, a=-3, c=0; c=-8, a=1, b=0; a=1.5, c=0,b=-3; b= , a= , c Match the equations with the following: a) the equation has two roots, b) the equation has one root, c) the equation has no roots. (c) (a) (b) (a) (a) (a) Match the equations with the following statements:

Check solution #515 (a, c, d). a). 4x 2 -9 \u003d 0 c). -0.1x2 +10=0 d). 6 v 2 +24 \u003d 0 4x 2 \u003d 9 -0.1x 2 \u003d - 10 6 v 2 \u003d -24 x 2 \u003d 9 / 4 x 2 \u003d - 10 / (-0.1) v 2 \u003d -24/6 x 1 \u003d -3 / 2 \u003d -1.5; x 2 \u003d 100 v 2 \u003d -4 x 2 \u003d 3/2 \u003d 1.5; x 1 = -10 Answer: no solution. Answer: -1.5; 1.5; Answer: -10;10;

04/28/17 Consider the solution of incomplete quadratic equations No. 517 (b, d, e) b). -5x2 + 6x=0 g). 4a 2 - 3a=0 e). 6 z 2 - z \u003d 0 x (-5x + 6) \u003d 0 a (4a-3) \u003d 0 z (6 z -1) \u003d 0 x \u003d 0 or -5x + 6 \u003d 0 a \u003d 0 or 4a-3 \u003d 0 z \u003d 0 or 6 z -1 \u003d 0 -5x \u003d -6 4a \u003d 3 6 z \u003d 1 x \u003d -6 / (-5) \u003d 1.2 a \u003d 3/4 \u003d 0.75 z \u003d 1 / 6 Answer: 0; 12. Answer: 0; 0.75. Answer: 0; 1/6...

1) For what values ​​of a is the equation a quadratic equation? No solutions 2) For what values ​​of a is the equation an incomplete quadratic equation?

3) Solve the equation for the obtained values ​​of a. Answer: a \u003d - 2, x \u003d - 15, x \u003d 0; a = 0,

Summing up What is the quadratic equation? Why a≠ 0 ? What are the names of the numbers a, b and c? How many types of incomplete quadratic equations have we learned? How are type I equations solved? Type II? Type III?

This is where our lesson ends. Guys! Did you receive answers to your questions? We realized that interesting things are ahead of us, and most importantly - important topics? I just want to remind you that when solving problems, examples, you need to look for rational approaches and apply a variety of methods.

Homework: Item 21 of the textbook; Nos. 318, 321 a, c, 323 a. Additionally: 520, 532. P. 21 (definitions), No. 518, 520 (a, c) 511 Additionally (for students with increased interest) No. 520, No. 531.


slide 2

"I have to divide my time between politics and equations. However, I think equations are much more important, because politics exists only for this moment, and equations will exist forever." A. Einstein.

slide 3

Hello guys!

Let's repeat: I am your assistant, I will guide you through the whole big topic of Quadratic Equations. In 7th and 8th grade, you already considered and even solved quadratic equations.

slide 4

Today you will learn: 1. What equations are called quadratic? 2. What is the main thing in the definition of a quadratic equation that should be remembered and taken into account? 3. What special cases of quadratic equations are there? 4. What are the ways to solve quadratic equations in each particular case? Now let's look for answers to these questions together. Good luck!

slide 5

What do these equations have in common?

slide 6

A quadratic equation is an equation of the form ... ax² + bx + c = 0, where a ≠ 0, x is a variable, a, b, c are some numbers. a is the senior (first) coefficient, b is the second coefficient, c is a free term. a is the senior (first) coefficient, b is the second coefficient, c is a free term. a is the senior (first) coefficient, c is the second coefficient, c is the free term.

Slide 7

If a = 1, then the quadratic equation x² + bx + c= 0 is called reduced. We will solve No. 513 (orally).

Slide 8

Let's try to solve:

5 5 -3 3 2 -4 1 4 3 -2 1 -1 4 -4 1

Slide 9

I wonder what will happen if the coefficients of the quadratic equation in turn or all at once (except a) turn into zeros. Let's do some research.

Slide 10

Incomplete quadratic equations

01/10/2017 10 If с=0, ax2+ bх= 0 ax2 ax2 If b,с=0, ax2= 0 If b=0, ax2+ c = 0

slide 11

Consider all possible cases

slide 12

slide 13

Incomplete quadratic equations of the form: no roots.

Slide 14

Incomplete quadratic equations of the form:

slide 15

Answer: x=0. no roots. Write incomplete quadratic equations:

slide 16

Write down the quadratic equations with the indicated coefficients: a=1, b=0, c=16; a=-1, b=5, c=0; b=0, a=-3, c=0; c=-8, a=1, b=0; a=1.5, c=0,b=-3; b= , a= , c Match the equations with the following: a) the equation has two roots, b) the equation has one root, c) the equation has no roots. (c) (a) (b) (a) (a) (a) Match the equations with the following statements:

Slide 17

17 Check solution No. 515 (a, c, d). a).4x2-9=0 c). -0.1x2+10=0 d). 6v2+24=0 4x2=9 -0.1x2=-10 6v2=-24x2=9/4x2=-10/(-0.1)v2=-24/6x1=-3/2=-1, five; x2=100 v2=-4 x2=3/2=1.5; x1=-10 Answer: no solution. Answer: -1.5; 1.5; Answer: -10; 10;

Slide 18

01/10/2017 18 Consider the solution of incomplete quadratic equations No. 517 (b, d, e) b). -5x2+ 6x=0 g). 4a2 - 3a=0e). 6z2–z =0 x(-5x+6)=0 a(4a-3)=0 z(6z –1) =0 x=0 or -5x+6=0 a=0 or 4a-3=0 z =0 or 6z –1 =0 -5x= -6 4a=36z=1 x = -6/(-5) =1.2 a=3/4=0.75 z=1/6 Answer: 0; 1.2. Answer: 0; 0.75. Answer: 0; 1/6...

Incomplete quadratic equations

Teacher of mathematics and physics: Balakina E.N.


LESSON OBJECTIVES:

  • Get to know the concept of a quadratic equation;
  • Learn to determine if an equation is quadratic; Learn to determine the coefficients of a quadratic equation; Compose a quadratic equation according to the given coefficients; Learn to determine the type of quadratic equation: complete or incomplete; Learn to choose an algorithm for solving an incomplete quadratic equation.
  • Learn to determine if an equation is quadratic;
  • Learn to determine the coefficients of a quadratic equation;
  • Compose a quadratic equation according to the given coefficients;
  • Learn to determine the type of quadratic equation: complete or incomplete;
  • Learn to choose an algorithm for solving an incomplete quadratic equation.

QUESTIONS:

  • What is an equation?
  • What does it mean to solve an equation?
  • What is the root of the equation?
  • What equations do we know?

Choose quadratic equations:

5x + 26 = 8x - 3,

  • + 22x - 2 = 0,

- 13x = 0,

  • - 53x +12 = 0,

9x + 2 - 17 = 0,

  • - 8 = 3,

34 + 5 - 22x = 11

9x + 7 - 13 = 0,

- 42x - 29 = 0,

-3 - 35x + 14 = 0,

+22 - 5x = 0,

-7 - 46x + 17 = 0,

8x - 6 = 0,

25 - 4x - 9 = 0.


THE QUADRATIVE EQUATION IS CALLED

EQUATION OF THE VIEW

a + bx+c=0,

where X - variable,

a,b,c- some numbers

and a = 0.

a is the first coefficient,

b is the second coefficient,

c is a free term.


Write a quadratic equation

  • - 7x + 12 = 0

-9 + 23x - 11 = 0

  • - 22x - 3 = 0

-4 + x + 5 = 0

4 + 9x = 0

+ 7x + 1 = 0

-3 + 15 = 0

-3 - x + 7 = 0

4 + 3 = 0

a=3, b=-7, c=12

a=-9, b=23, c=-11

a=8, b=0, c=0

a=5, b=-22, c=-3

a=-4, b=1, c=5

a=4, b=9, c=0

a=1, b=7, c=1

a=-3, b=0, c=15

a=-3, b=-1, c=7

a=4, b=0, c=3


If in a quadratic equation a + bx+c=0 at least one of the coefficients b or from is equal to zero, then such an equation is called incomplete quadratic equation.

There are three types of incomplete quadratic equations:

  • a = 0
  • a +b x = 0
  • a + c = 0

1 option

- ; At 0;3 AND 0;-2 P n.r. IN -3;3 R 0;2 E 0 H 0;4 BUT -2,5;2,5 ABOUT - ; D

Option 2

  • + 2x = 0
  • 2 - 18 = 0
  • 4 - 11= - 11+ 9x
  • 9 + 1 = 0
  • 2 = 4x
  • 7 - 14 = 0
  • 9 - 2 + 16x = 6 + 9
  • - 4 = 0
  • 9 + 1 = 1
  • 4 - 25 = 0
  • -2 + 4x = 0
  • - 3x = 0
  • 7 = 0
  • 12x = 6
  • 2 = 7 + 2
  • 6 + 24 = 0
  • 3 + 7 = 12x + 7
  • + 2x - 3 = 2x + 6
  • 9 - 4 = 0
  • 7x = 2 + 3x

The numbers are written on the board 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1

Pupils write out the letters corresponding to the roots of these equations; options work towards each other.


REFERRED QUADRATIVE EQUATION

Called square

equation in which the coefficient

when equals 1:

+ bx+c=0


HOMEWORK

№ 24.11 (ORAL),

№ 24.16 (b, c, d),

№ 24.18 (b, c, d).


History reference

Quadratic equations were solved in Babylon around 2000 BC.

In Europe, in 2002, they celebrated the 800th anniversary of quadratic equations, because it was in 1202 that the Italian scientist Leonard Fibonacci laid out the formulas for a quadratic equation.

Only in the 17th century, thanks to Newton, Descartes and other scientists, did these formulas take on a modern form.


IN ancient india already in 499 public competitions were distributed in solving problems for the compilation of quadratic equations. One of these problems is the problem of the famous Indian mathematician Bhaskara :

Frisky flock of monkeys Eating well, having fun, Them squared part eight Having fun in the meadow. And twelve by vines They began to jump, hanging. How many monkeys were You tell me in this flock?

 

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