9x2−9(a+b)x+2a2+5ab+2b2=0cari nilai x

9×2−9(a+b)x+2a2+5ab+2b2=0cari nilai x

Jawab:

$x={bfdfrac{2a+b}{3}} {sf atau} x={bfdfrac{a+2b}{3}}$

PEMBAHASAN

Persamaan/Fungsi Kuadrat

$begin{aligned}&9x^2-9(a+b)x+2a^2+5ab+2b^2=0\Bigl.&quadleft[ textsf{bagi kedua ruas dengan 9}right.\Biggl.&{Leftrightarrow }x^2-(a+b)x+frac{2a^2+5ab+2b^2}{9}=0\Bigl.&quadleft[ textsf{faktorkan $2a^2+5ab+2b^2$}right.\Biggl.&{Leftrightarrow }x^2-(a+b)x+frac{(2a+b)(a+2b)}{9}=0\Biggl.&{Leftrightarrow }x^2-(a+b)x+frac{2a+b}{3}cdotfrac{a+2b}{3}=0end{aligned}$

$begin{aligned}Bigl.&quadleft[ begin{aligned}Biggl.&{sf Ingat:} (x-m)(x-n)=x^2-(m+n)x+mn\Biggl.&textsf{Jika $m=frac{2a+b}{3}$ dan $n=frac{a+2b}{3}$, maka:}\Biggl.&quad m+n=frac{2a+b}{3}+frac{a+2b}{3}\Biggl.&qquadqquad!!=frac{3a+3b}{3} = a+b\Biggl.&quad mn=frac{2a+b}{3}cdotfrac{a+2b}{3}end{aligned}right.\Biggl.&{Leftrightarrow }left(x-frac{2a+b}{3}right)left(x-frac{a+2b}{3}right)=0end{aligned}$

$begin{aligned}Biggl.&{thereforequad}x={bffrac{2a+b}{3}} {sf atau} x={bffrac{a+2b}{3}}end{aligned}$