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Задать вопрос экспертам

Консультация № 177242

14.03.2010, 03:01

0.00 руб.

0 3 2

Образование

Здравствуйте=)

При каких значениях а система уравнений имеет единственное решение?
y = [$8730$](x+1)
y=a+x

Заранее благодарю=)

Обсуждение

давно

Асмик Гаряка

Профессор
230118
3054

14.03.2010, 11:33

общий

это ответ

Здравствуйте, Schuldig.

y=[$8730$]x+1 - ветвь параболы.
y=x+a - прямая. Они могут пересекаться в 2 точках, касаться или не пересекаться в зависимости от значения a.
Точка касания определяется нулем дискриминанта.
(a+x)²=x+1
x²+(2a-1)x+a²-1=0
D=(2a-1)²-4(a²-1)=4a²-4a+1-4a²+4=5-4a=0
a=1.25
x²+3/2x+9/16=0
(x+3/4)²=0
x=-3/4
y=1/2
При a<1.25 квадратное уравнение имеет 2 корня, но надо проверить на область определения.
x₁=(-(2a-1)-[$8730$]D)/2)>-1
-(2a-1)-[$8730$]D)/2>-2
-2a+3>[$8730$]5-4a
(-2a+3)²>5-4a, a<1.25
4a²-12a+9>5-4a
4a²-8a+4>0
4(a-1)²>0
y>0
y₁=(-(2a-1)-[$8730$]D)/2)/2+a>0
-2a+1-[$8730$]D+2a>0
1-[$8730$]D>0
[$8730$]D<1
5-4a<1
4a>4
a>1
y₂=(-(2a-1)+[$8730$]D)/2)/2+a>0
1+[$8730$]D>0 выполняется всегда при a<1.25
При 1<a<5/4 имеется 2 корня.
При a>5/4 корней нет.
Ответ: (-[$8734$],1)[$8746$]{1.25}

5

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

14.03.2010, 18:27

общий

это ответ

Здравствуйте, Schuldig.

5

давно

Лысков Игорь Витальевич

Посетитель
7438
7205

16.03.2010, 02:34

общий

Гаряка Асмик:
Возведение в квадрат добавило лишний корень...

Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен

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