Question

QUIZZZ !!!

[tex] \frac{ {x}^{2} }{x + 6} + \frac{x}{ {x}^{2} - 6 } = 2[/tex]
Jika x≠ ±√6 , x ≠ 0 , dan x≠ -6

maka nilai x {....,....,....,....} ??​

Answer 1

[tex] \frac{ {x}^{2} }{x + 6} + \frac{x}{ {x}^{2} - 6 } = 2[/tex]

[tex] \frac{ {x}^{4} - 6 {x}^{2} + {x}^{2} + 6x}{x {}^{3} - 6x + 6 {x}^{2} - 36 } = 2[/tex]

[tex] {x}^{4} - {5x}^{2} + 6x = {2x}^{3} - 12x + {12x}^{2} - 72[/tex]

[tex] {x}^{4} - {2x}^{3} - {17x}^{2} + 18x + 72 = 0[/tex]

[tex](x - 4)( {x}^{3} + {2x}^{2} - 9x - 18) = 0[/tex]

[tex](x - 4)(x - 3)( {x}^{2} + 5x + 6) = 0[/tex]

[tex](x - 4)(x - 3)(x + 2)(x + 3) = 0[/tex]

x={4,3,-3,-2}

Answer 2

Verified answer

Nilai x yang memenuhi adalah x ∈ {–2, –3, 3, 4}.
 

Penjelasan dengan langkah-langkah

[tex]\displaystyle\frac{x^2}{x+6}+\frac{x}{x^2-6}=2[/tex]

Samakan penyebut.

[tex]\displaystyle\frac{x^2\left(x^2-6\right)+x(x+6)}{(x+6)\left(x^2-6\right)}=2[/tex]

Kali silang dan lanjutkan.

[tex]\begin{aligned}2(x+6)\left(x^2-6\right)&=x^2\left(x^2-6\right)+x(x+6)\\(x+6)\left(x^2-6\right)+(x+6)\left(x^2-6\right)&=x^2\left(x^2-6\right)+x(x+6)\\(x+6)\left(x^2-6\right)-x(x+6)&=x^2\left(x^2-6\right)-(x+6)\left(x^2-6\right)\\(x+6)\left(x^2-6-x\right)&=\left(x^2-6\right)\left(x^2-x-6\right)\end{aligned}[/tex]

Dari kesamaan kedua ruas, nilai x yang memenuhi adalah akar-akar dari x² – x – 6 = 0 atau x² – 6 = x + 6

• Untuk x² – x – 6 = 0:
  (x + 2)(x – 3) = 0
  ⇔ x = –2, x = 3
• Untuk x² – 6 = x + 6:
  x² – 6 – x – 6 = 0
  ⇔ x² – x – 12 = 0
  ⇔ (x + 3)(x – 4) = 0
  ⇔ x = –3, x = 4

∴ Jadi, x ∈ {–2, –3, 3, 4}.