نویسنده : امیر انصاری

1. $$
   x^2-8x-4=0
   $$
   
2. $$
   -3x^2+4x+5=0
   $$
   
3. $$
   \frac{1}{2}x^2-6x-5=0
   $$
   
4. $$
   0.2x^2+0.12x-11=0
   $$
   
5. $$
   -\frac{2}{3}x^2-x+2=0
   $$
   
6. $$
   \frac{3}{4}x^2+6x+1=0
   $$
   

پاسخ

1. $$
   x^2-8x-4=0\\
   x^2-8x=4\\
   x^2-8x+16=4+16\\
   (x-4)^2=20\\
   x-4=\pm \sqrt{20}\\
   \text{ }\\[2ex]
   x-4=\sqrt{20}\\
   x=4+\sqrt{20}\\
   x \approx 8.5\\
   \text{ }\\[2ex]
   x-7=-\sqrt{20}\\
   x=4-\sqrt{20}\\
   x \approx -0.5
   $$
   
2. $$
   -3x^2+4x+5=0\\
   x^2-\frac{4}{3}x=\frac{5}{3}\\
   x^2-\frac{4}{3}x+\frac{4}{9}=\frac{5}{3}+\frac{4}{9}\\
   \bigl( x-\frac{2}{3} \bigr)^2 = \frac{19}{9}\\
   x - \frac{2}{3} = \pm \sqrt{\frac{19}{9}}\\
   x = \frac{2 \pm \sqrt{19}}{3}\\
   \text{ }\\[2ex]
   x = \frac{2+\sqrt{19}}{3}\\
   x \approx 2.1\\
   \text{ }\\[2ex]
   x=\frac{2-\sqrt{19}}{3}\\
   x \approx -0.8
   $$
   
3. $$
   \frac{1}{2}x^2-6x-5=0\\
   x^2-12x=10\\
   x^2-12x+36=10+36\\
   (x-6)^2=46\\
   x-6=\pm\sqrt{46}\\
   \text{ }\\[2ex]
   x-6=\sqrt{46}\\
   x=6+\sqrt{46}\\
   x \approx 12.8\\
   \text{ }\\[2ex]
   x-6 = -\sqrt{46}\\
   x = 6-\sqrt{46}\\
   x \approx -0.8
   $$
   
4. $$
   0.2x^2+0.12x-11=0\\
   x^2+0.6x=55\\
   x^2+0.6x+0.09=55+0.09\\
   (x+0.3)^2=55.09\\
   x+0.3=\pm\sqrt{55.09}\\
   \text{ }\\[2ex]
   x+0.3 = \sqrt{55.09}\\
   x=-0.3 + \sqrt{55.09}\\
   x \approx 7.1\\
   \text{ }\\[2ex]
   x+0.3=-\sqrt{55.09}\\
   x=-0.3-\sqrt{55.09}\\
   x \approx -7.7
   $$
   
5. $$
   -\frac{2}{3}x^2-x+2=0\\
   x^2+\frac{3}{2}x=3\\
   x^2+\frac{3}{2}x+\frac{9}{16}=3+\frac{9}{16}\\
   \bigl( x+\frac{3}{4} \bigr)^2=\frac{57}{16}\\
   x+\frac{3}{4}=\pm \sqrt{\frac{57}{16}}\\
   x=\frac{-3 \pm \sqrt{57}}{4}\\
   \text{ }\\[2ex]
   x=\frac{-3+\sqrt{57}}{4}\\
   x \approx 1.1\\
   \text{ }\\[2ex]
   x=\frac{-3-\sqrt{57}}{4}\\
   x \approx -2.6
   $$
   
6. $$
   \frac{3}{4}x^2+6x+1=0\\
   x^2+8x=-\frac{4}{3}\\
   x^2+8x+16=-\frac{4}{3}+16\\
   (x+4)^2=\frac{44}{3}\\
   x+4=\pm\sqrt{\frac{44}{3}}\\
   \text{ }\\[2ex]
   x+4=\sqrt{\frac{44}{3}}\\
   x=-4 + \sqrt{\frac{44}{3}}\\
   x \approx -0.2\\
   \text{ }\\[2ex]
   x+4=-\sqrt{\frac{44}{3}}\\
   x=-4 - \sqrt{\frac{44}{3}}\\
   x \approx -7.8
   $$