Skorzystamy ze wzoru skróconego mnożenia na różnicę kwadratów: $a^2-b^2=\left(a-b\right)\left(a+b\right)$

---

$\text{a)}{\left(x-1\right)}^2-9=0$  

${\left(x-1\right)}^2-3^2=0$  

$\left(x-1-3\right)\left(x-1+3\right)=0$  

$\left(x-4\right)\left(x+2\right)=0$  

$x-4=0\text{lub}x+2=0$  

$x=4\text{lub}x=-2$  

---

$\text{b)}{\left(3x-1\right)}^2-25=0$  

${\left(3x-1\right)}^2-5^2=0$  

$\left(3x-1-5\right)\left(3x-1+5\right)=0$  

$\left(3x-6\right)\left(3x+4\right)=0$  

$3x-6=0\text{lub}3x+4=0$  

$3x=6\text{lub}3x=-4\mid :3$  

$x=2\text{lub}x=-\frac{4}{3}$  

---

$\text{c)}{\left(5+x\right)}^2-5=0$  

${\left(x+5\right)}^2-{\left(\sqrt{5}\right)}^2=0$  

$\left(x+5-\sqrt{5}\right)\left(x+5+\sqrt{5}\right)=0$  

$x+5-\sqrt{5}=0\text{lub}x+5+\sqrt{5}=0$  

$x=-5+\sqrt{5}\text{lub}x=-5-\sqrt{5}$  

---

$\text{d)}{\left(8x-1\right)}^2-{\left(8x+1\right)}^2=0$  

$\left[\left(8x-1\right)-\left(8x+1\right)\right]\left[\left(8x-1\right)+\left(8x+1\right)\right]=0$  

$\left(8x-1-8x-1\right)\left(8x-1+8x+1\right)=0$  

$-2\cdot 16x=0$  

$-32x=0\mid :\left(-32\right)$  

$x=0$  

---

$\text{e)}{\left(\frac{2}{3}x-1\right)}^2-x^2=0$  

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

$\left(-\frac{1}{3}x-1\right)\left(1\frac{2}{3}x-1\right)=0$  

$-\frac{1}{3}x-1=0\text{lub}1\frac{2}{3}x-1=0$  

$\frac{1}{3}x=-1\left|\cdot 3\text{lub}\frac{5}{3}x=1\right|\cdot \frac{3}{5}$  

$x=-3\text{lub}x=\frac{3}{5}$  

---

$\text{f)}4{\left(x-1\right)}^2-9{\left(x+2\right)}^2=0$  

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

${\left[2\left(x-1\right)\right]}^2-{\left[3\left(x+2\right)\right]}^2=0$  

$\left[2\left(x-1\right)-3\left(x+2\right)\right]\left[2\left(x-1\right)+3\left(x+2\right)\right]=0$  

$\left(2x-2-3x-6\right)\left(2x-2+3x+6\right)=0$  

$\left(-x-8\right)\left(5x+4\right)=0$  

$-x-8=0\text{lub}5x+4=0$  

$x=-8\text{lub}5x=-4\mid :5$  

$x=-8\text{lub}x=-\frac{4}{5}$