persamaan kuadrat x²+ax+5a+3=0 mempunyai dua akar real alpa dan beta. jika alpa+2beta=3 maka konstanta a adalah

Mapel : Matematika
Kelas : X SMA
Bab : Persamaan Kuadrat

Pembahasan :
x² + ax + (5a + 3) = 0

Berdasarkan Teorema Vietta
a + β = -b/a
a + β = -a

aβ = c/a
aβ = 5a + 3

Maka...
a + 2β = 3
a = 3 - 2β

Substitusi...
a + β = -a
3 - 2β + β = -a
β = 3 + a

aβ = 5a + 3
(-6 - 2a)(3 + a) = 5a + 3
-18 - 6a - 6a - 2a² = 5a + 3
-2a² - 12a - 18 = 5a + 3
-2a² - 17a - 21 = 0
2a² + 17a + 21 = 0
(2a + 3)(a + 7) = 0
a = -3/2
a = -7

x² + ax + 5a + 3 = 0
α + 2β = 3
α + β + β = 3
(-a/1) + β = 3
β = a + 3

x² + ax + 5a + 3 = 0
(a + 3)² + a(a + 3) + 5a + 3 = 0
a² + 6a + 9 + a² + 3a + 5a + 3 = 0
2a² + 14a + 12 = 0
a² + 7a + 6 = 0
(a + 6)(a + 1) = 0
a = -6 atau a = -1