Quadratic equations location of roots.

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Документ обсуждает решение квадратного уравнения x2 - 2(b-1)x + 2b + 1 = 0 и определяет условия для кол-ва корней в зависимости от параметра b. Рассматриваются случаи с различными значениями b, включая условия, при которых один или оба корня равны 0, их знаки и количество. Также приводится список литературы по алгебре.

Education

1) Решить уравнение.
X2-2(b-1)x+2b+1=0
D/4=(-(b-1))2 -2b-1=(1-2b+b2)-2b-
-1=b2-2b+1-2b-1=b2-4b=b(b-4).
1) Если D/4<0, то есть b(b-4)<0
b < 0,
 b < 0,
b − 4 > 0; 
b > 4;
⇔ b > 0, ⇔ b > 0, ⇔ 0 < b < 4.
 
b − 4 < 0 b < 4

b = 0,
2) Если D/4=0, то есть b(b-4)=0 <=> 
то уравнение имеет 1 корень x= b = 4.
= -b/2a, то есть х=а-1(а=0 х=-1; а=4 х=3)
3) Если D/4>0 то есть b(b-4)>0 <=>
b > 0, b > 0,
 
b − 4 > 0; b > 4; b > 4,
⇔ b < 0,
⇔ b < 0, ⇔ b < 0.
  
b − 4 < 0 b < 4;
то уравнение имеет 2 корня х1,2= b − 1 ± b2 − 4b
Ответ: если 0<b<4, то решений нет;
если b=0, то х=-1; если b=4, то х=3;
если b ∈ ( −∞;0) ∪ (4;+∞) , то х = b − 1 ± b2 − 4b
1,2

Исследовать

2) При каких значениях b
ровно 1 из корней
уравнения равен 0?

Ровно 1 из корней уравнения равен 0
тогда и только тогда, когда: c = 0,

b ≠ 0.
Получим: 2b + 1 = 0, b = −1 / 2,
⇔ ⇒ b = −1 / 2

− 2(b − 1) ≠ 0 b ≠ 1
Ответ: b = −1 / 2.

3)При каких значениях b
корни уравнения равны по
модулю и противо-
полжны по знаку?
Корни уравнения равны по модулю и
противоположны по знаку, тогда и только
тогда, когда: b = 0,

 − c / a ≠ 0.
Получим: − 2( a − 1) = 0, a = 1, - решений нет.
 ⇔
 − 2a − 1 > 0 a < −1 / 2.

Ответ: решений нет.

4) При каких значениях b
корни уравнения равны
0?

Оба корня уравнения равны нулю, тогда
и только тогда, когда: b = 0,

 c = 0.
Получим: − 2(b − 1) = 0, b = −1 / 2,
 ⇔
2b + 1 = 0 b = 1.

Ответ: b=1и b=-1/2.

5) При каких значениях b
корни уравнения
положительны?
 D / 4 > 0,

− b / a < 0, - необходимое и достаточное условие.
c / a > 0.
  b ≥ 4,
Получим: b(b − 4) ≥ 0, 
  b ≤ 0;

2(b − 1) > 0, ⇔ b > 1, ⇒ b ≥ 4.
2 a + 1 > 0 b > −1 / 2





Ответ: b ≥ 4.

6) При каких значениях
b корни уравнения
отрицательны?

 D / 4 > 0,
 - необходимое и достаточное условие.
− b / a < 0,
 c / a > 0.
  b = 0,

b(b − 4) > 0,
Получим:   b ≥ 4;

2(b − 1) < 0, ⇔ b < 1, ⇒ − 1 / 2 < b < 0.
 2b + 1 > 0 b > −1 / 2
 



Ответ: b ∈ (−1 / 2;0)

7) При каких значениях b
корни уравнения
одного знака?

 D / 4 ≥ 0,
 - необходимое и достаточное условие.
c / a > 0.

Получим: b 2 − 4b ≥ 0,  b ≥ 4,
 − 1 / 2 < b < 0,
 ⇔  b ≤ 0; ⇔ 
 2b + 1 > 0 b ≥ −1 / 2  b ≥ 4.


Ответ: b ∈ ( −1 / 2;0] ∪ [4;+∞)

При каких значениях b
корни уравнения
разных знаков?

c / a < 0. - необходимое и достаточное условие.

Получим: 2b+1<0,
2b<-1,
b<-1/2.

Ответ: b ∈ ( −∞;−1 / 2).

Список литературы.
Учебник по алгебре, 8-9/ М. Л. Галицкий,
А. М. Гольдман, Л. И. Звавич.
«Просвещение»
Учебник по алгебре ,8 кл.
С. М. Никольский, М. К. Потапов,
Н. Н. Решетников, А. В. Шевкин.
«Просвещение»

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Quadratic equations location of roots.

2. 1) Решить уравнение. X2-2(b-1)x+2b+1=0 D/4=(-(b-1))2 -2b-1=(1-2b+b2)-2b- -1=b2-2b+1-2b-1=b2-4b=b(b-4). 1) Если D/4<0, то есть b(b-4)<0 b < 0,  b < 0, b − 4 > 0;  b > 4; ⇔ b > 0, ⇔ b > 0, ⇔ 0 < b < 4.   b − 4 < 0 b < 4

3. b = 0, 2) Если D/4=0, то есть b(b-4)=0 <=>  то уравнение имеет 1 корень x= b = 4. = -b/2a, то есть х=а-1(а=0 х=-1; а=4 х=3) 3) Если D/4>0 то есть b(b-4)>0 <=> b > 0, b > 0,   b − 4 > 0; b > 4; b > 4, ⇔ b < 0, ⇔ b < 0, ⇔ b < 0.    b − 4 < 0 b < 4; то уравнение имеет 2 корня х1,2= b − 1 ± b2 − 4b Ответ: если 0<b<4, то решений нет; если b=0, то х=-1; если b=4, то х=3; если b ∈ ( −∞;0) ∪ (4;+∞) , то х = b − 1 ± b2 − 4b 1,2

4. Исследовать 2) При каких значениях b ровно 1 из корней уравнения равен 0? Ровно 1 из корней уравнения равен 0 тогда и только тогда, когда: c = 0,  b ≠ 0. Получим: 2b + 1 = 0, b = −1 / 2, ⇔ ⇒ b = −1 / 2  − 2(b − 1) ≠ 0 b ≠ 1 Ответ: b = −1 / 2.

5. 3)При каких значениях b корни уравнения равны по модулю и противо- полжны по знаку? Корни уравнения равны по модулю и противоположны по знаку, тогда и только тогда, когда: b = 0,   − c / a ≠ 0. Получим: − 2( a − 1) = 0, a = 1, - решений нет.  ⇔  − 2a − 1 > 0 a < −1 / 2. Ответ: решений нет.

6. 4) При каких значениях b корни уравнения равны 0? Оба корня уравнения равны нулю, тогда и только тогда, когда: b = 0,   c = 0. Получим: − 2(b − 1) = 0, b = −1 / 2,  ⇔ 2b + 1 = 0 b = 1. Ответ: b=1и b=-1/2.

7. 5) При каких значениях b корни уравнения положительны?  D / 4 > 0,  − b / a < 0, - необходимое и достаточное условие. c / a > 0.   b ≥ 4, Получим: b(b − 4) ≥ 0,    b ≤ 0;  2(b − 1) > 0, ⇔ b > 1, ⇒ b ≥ 4. 2 a + 1 > 0 b > −1 / 2     Ответ: b ≥ 4.

8. 6) При каких значениях b корни уравнения отрицательны?  D / 4 > 0,  - необходимое и достаточное условие. − b / a < 0,  c / a > 0.   b = 0,  b(b − 4) > 0, Получим:   b ≥ 4;  2(b − 1) < 0, ⇔ b < 1, ⇒ − 1 / 2 < b < 0.  2b + 1 > 0 b > −1 / 2     Ответ: b ∈ (−1 / 2;0)

9. 7) При каких значениях b корни уравнения одного знака?  D / 4 ≥ 0,  - необходимое и достаточное условие. c / a > 0. Получим: b 2 − 4b ≥ 0,  b ≥ 4,  − 1 / 2 < b < 0,  ⇔  b ≤ 0; ⇔   2b + 1 > 0 b ≥ −1 / 2  b ≥ 4.  Ответ: b ∈ ( −1 / 2;0] ∪ [4;+∞)

10. При каких значениях b корни уравнения разных знаков? c / a < 0. - необходимое и достаточное условие. Получим: 2b+1<0, 2b<-1, b<-1/2. Ответ: b ∈ ( −∞;−1 / 2).

11. Список литературы. Учебник по алгебре, 8-9/ М. Л. Галицкий, А. М. Гольдман, Л. И. Звавич. «Просвещение» Учебник по алгебре ,8 кл. С. М. Никольский, М. К. Потапов, Н. Н. Решетников, А. В. Шевкин. «Просвещение»

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Quadratic equations location of roots.

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