2.2

231

a)
$\frac{x}{x+1}=\frac{3x}{2x+1}$
$x\left(2x+1\right)=3x\left(x+1\right)$
$2x^2+x-3x^2-3x=0$
$-x^2-2x=0$
$-x\left(x+2\right)=0$
$x=0$
$x+2=0$
$x=-2$
b)
$\frac{2x}{1+x}=\frac{x}{1-x}$
$2x\left(1-x\right)=x\left(1+x\right)$
$2x-2x^2-x-x^2$
$-3x^2+x=0$
$x\left(-3x+1\right)=0$
$x=0$
$-3x+1=0$
$3x=1$
$x=\frac{1}{3}$
c)
$\frac{x}{x+1}=\frac{x+2}{x}$
$x^2=\left(x+1\right)\left(x+2\right)$
$x^2=x^2+2x+x+2$
$x^2-x^2-3x-2$
$-3x-2=0$
$x+\frac{2}{3}=0$
$x=-\frac{2}{3}$

230

a)
$3\left(x-1\right)\left(x-7\right)=0$
$\left(3x-3\right)\left(x-7\right)=0$
$3x-3=0$
$x=1$
$x-7=0$
$x=7$
nollakohdat x=1 ja x=7
b) 1<x<7

224

a) 3
nollakohdat on x=0 ja x=-2
b) 2
nollakohdat on x=0 ja x=2
c) 1
nollakohdat on x=-2 ja x=2

234

a)
nollakohdat:
$x^2-8x=0$
$x\left(x-8\right)=0$
$x=0$
$x=8$
huipun x:
$\frac{0+8}{2}=4$
huipun y:
$4^2-8\cdot4=16-32=-16$
huipun koordinaatit: (4, -16)
b)
samalla tavoin kuin a-kohta huipun y asti:
nollakohdat:
$x^2-8x=0$
$x\left(x-8\right)=0$
$x=0$
$x=8$
huipun x:
$\frac{0+8}{2}=4$
huipun y:
$4^2-8\cdot4+7=16-32+7=-9$
huipun koordinaatit (4, -9)

230

a)
$3\left(x-1\right)\left(x-7\right)=0$
$\left(3x-3\right)\left(x-7\right)=0$
nollakohdat:
$3x-3=0$
$3x=3$
$x=1$
tai
$x-7=0$
$x=7$
v: x=1 tai x=7

b) 1<x<7

228

a) x=3
$f\left(x\right)=\left(x+3\right)$
$x+3=0$
$x=-3$
x-3=0
b) x=2 tai x=-5
$f\left(x\right)=\left(x-2\right)\left(x+5\right)$ =0

227

a)
$f\left(x\right)=\left(x+2\right)\left(x-3\right)$
$x^2-3x+2x-6$
$x^2-x-6$
b)
nollakohdat x=-1 ja x=4
$f\left(x\right)=\left(x+1\right)\left(x-4\right)$
$x^2-4x+x-4$
$x^2-3x-4$
c)

221

a) 2x+4
=2(x+2)
b)
$2x^2+4x$
$2x\left(x+2\right)$
c)
$6x^2-9x$
$3x\left(2x-3\right)$