gui ban Thanh ve bai toan tim m de P(x) = (x + m)/6 co 3 nghiem phan biet?

Cac ban da hieu nham do khong phai la pt bac 2 nen cach giai nhu sau

a) P(x) = √[x + 6√(x - 9)] + √[x - 6√(x - 9)]

= √[(x - 9) + 2.3.√(x - 9) + 9] + √[(x - 9) - 2.3√(x - 9) + 9]

= √[√(x - 9) + 3]² + √[√(x - 9) - 3]²

@ Neu x > 18 thi x - 9 > 9 => √(x - 9) - 3 > 0 => √[√(x - 9) - 3]² = √(x - 9) - 3

=> P(x) = √(x - 9) + 3 + √(x - 9) - 3 = 2√(x - 9)

@ Neu 9 =< x =< 18 thi 0 =< x - 9 =< 9 => √(x - 9) - 3 =< 0 => √[√(x - 9) - 3]² = 3 - √(x - 9)

=> P(x) = √(x - 9) + 3 + 3 - √(x - 9) = 6

b) Theo cau a) thi

P(x) = (x + m)/6 = 6 neu 9 =< x =< 18 (1)

P(x) = (x + m)/6 = 2√(x - 9) neu x > 18 (2)

Vi (1) co toi da 1 nghiem va (2) co toi da 3 nghiem nen de pt co 3 nghiem pb thi (1) phai co 1 nghiem va (2) co 2 nghiem pb

@ Tu (1) => x = 36 - m => 9 < 36 - m =< 18 => 18 =< m =< 27 (3)

@ Tu (2) va (3) => x + m = 12√(x - 9)

=> x² + 2mx + m² = 144(x - 9)

=> x² + 2(m - 72)x + m² + 1296 = 0

Delta' = (m - 72)² - (m² + 1296) = 144(27 - m) > 0 => m < 27 (4)

Ket hop (3) va (4) ta co 18 =< m < 27. (5)

Ta se cm rang voi dk (5) thi ca 2 nghiem x1; x2 deu > 18

That vay xet nghiem nho

x1 = (72 - m) - 12√(27 - m) = 9 + (27 - m) - 2.6√(27 - m) + 36

= 9 + [6 - √(27 - m)]² > 9 + [6 - √(27 - 18)]² > 18

=> x2 = (72 - m) + 12√(27 - m) > 18

KL : Voi 18 =< m < 27 thi pt co 3 nghiem phan biet