Question

Jika [tex]x_1[/tex] dan [tex]x_2[/tex] adalah akar-akar persamaan kuadrat 2x² - 5x + 6 = 0
Maka tentukan:
[tex](x_1 - {1})^{2} + (x_2 - {1})^{2}[/tex]
​

Answer 1

Verified answer

Jawaban:

-11/4

Penjelasan dengan langkah-langkah:

a=2

b=-5

c=6

[tex](x_1 - {1})^{2} + (x_2 - {1})^{2}[/tex]

= [tex](x_1^{2}-2x_1+1) + (x_2^{2}-2X_2+1)[/tex]

= [tex](x_1^{2}+x_2^{2})-2(x_1+x_2)+2[/tex]

= [tex](x_1+x_2)^{2}-2(x_1 x_2)-2(x_1+x_2)+2[/tex]

= [tex](-b/a)^{2}-2(c/a)-2(-b/a)+2[/tex]

= [tex](5/2)^{2}-2(3)-2(5/2)+2[/tex]

= [tex](25/4)-6-5+2[/tex]

= [tex](25/4)-(36/4)[/tex]

= -11/4

Answer 2

Teorema Vieta

[tex]\boxed{x_{1}+x_{2}=-\frac{b}{a} }\\\\\boxed{x_{1}\cdot x_{2}=\frac{c}{a} }\\\\\boxed{x_{1}^{2} +x_{2}^{2} =\frac{b^{2}-2ac }{a^{2} } }[/tex]

Penyelesaian:

2x² - 5x + 6 = 0

a = 2, b = -5, c = 6

x₁ + x₂ = -b/a = 5/2

x₁² + x₂² = (b² - 2ac)/a² = ((-5)² - 2(2)(6))/2² = (25 - 24)/4 = 1/4

(x₁ - 1)² + (x₂ - 1)²

= (x₁² - 2x₁ + 1) + (x₂² - 2x₂ + 1)

= (x₁² + x₂²) - 2(x₁ + x₂) + 2

= 1/4 - 2(5/2) + 2

= 1/4 - 10/2 + 3

= 1/4 - 20/4 + 12/4

= -11/4

Jadi, hasilnya adalah -11/4.