Complete the square:

\(x^2+2x-5=(x+1)^2-1-5=(x+1)^2-6\).

Complete the square:

\(x^2-10x+20=(x-5)^2-25+20=(x-5)^2-5\).

Complete the square:

\(x^2-4x+1=(x-2)^2-4+1=(x-2)^2-3\).

Factor out the negative sign from the quadratic part:

\(6-8x-x^2=-(x^2+8x)+6\).

Complete the square inside:

\(x^2+8x=(x+4)^2-16\).

So

\(6-8x-x^2=-[(x+4)^2-16]+6=22-(x+4)^2\).

\(10-16x-x^2=-(x^2+16x)+10\).

Now

\(x^2+16x=(x+8)^2-64\).

So

\(10-16x-x^2=-[(x+8)^2-64]+10=74-(x+8)^2\).

\(x^2+20x-45=(x+10)^2-100-45=(x+10)^2-145\).

\(x^2+12x+16=(x+6)^2-36+16=(x+6)^2-20\).

\(3-6x-x^2=-(x^2+6x)+3\).

\(x^2+6x=(x+3)^2-9\).

So

\(3-6x-x^2=-[(x+3)^2-9]+3=12-(x+3)^2\).

\(x^2+6x+1=0\)

\((x+3)^2-8=0\)

\((x+3)^2=8\)

\(x+3=\pm 2\sqrt{2}\)

\(x=-3\pm 2\sqrt{2}\).

\(x^2-4x-8=0\)

\((x-2)^2-12=0\)

\((x-2)^2=12\)

\(x-2=\pm 2\sqrt{3}\)

\(x=2\pm 2\sqrt{3}\).

Rearrange:

\(x^2+2x-4=0\).

\((x+1)^2-5=0\)

\((x+1)^2=5\)

\(x+1=\pm \sqrt{5}\)

\(x=-1\pm \sqrt{5}\).

\((x-10)^2-70=0\)

\((x-10)^2=70\)

\(x=10\pm \sqrt{70}\).

Rearrange:

\(x^2-8x-3=0\).

\((x-4)^2-19=0\)

\((x-4)^2=19\)

\(x=4\pm \sqrt{19}\).

\((x+7)^2-50=0\)

\((x+7)^2=50\)

\(x+7=\pm 5\sqrt{2}\)

\(x=-7\pm 5\sqrt{2}\).

\(x^2-3x-2=0\)

\(\left(x-\dfrac{3}{2}\right)^2-\dfrac{17}{4}=0\)

\(\left(x-\dfrac{3}{2}\right)^2=\dfrac{17}{4}\)

\(x-\dfrac{3}{2}=\pm \dfrac{\sqrt{17}}{2}\)

\(x=\dfrac{3\pm \sqrt{17}}{2}\).

\(x^2-x-3=0\)

\(\left(x-\dfrac12\right)^2-\dfrac{13}{4}=0\)

\(\left(x-\dfrac12\right)^2=\dfrac{13}{4}\)

\(x-\dfrac12=\pm \dfrac{\sqrt{13}}{2}\)

\(x=\dfrac{1\pm \sqrt{13}}{2}\).

Factor out 6 from the quadratic terms:

\(6x^2+12x-3=6(x^2+2x)-3\).

Complete the square inside:

\(x^2+2x=(x+1)^2-1\).

So

\(6[(x+1)^2-1]-3=6(x+1)^2-9\).

\(3x^2-6x-15=3(x^2-2x)-15\).

\(x^2-2x=(x-1)^2-1\).

So

\(3[(x-1)^2-1]-15=3(x-1)^2-18\).

\(3x^2-18x+4=3(x^2-6x)+4\).

\(x^2-6x=(x-3)^2-9\).

So

\(3[(x-3)^2-9]+4=3(x-3)^2-23\).

\(4x^2+24x-9=4(x^2+6x)-9\).

\(x^2+6x=(x+3)^2-9\).

So

\(4[(x+3)^2-9]-9=4(x+3)^2-45\).

Factor out \(-2\):

\(5-2x-2x^2=5-2(x^2+x)\).

\(x^2+x=\left(x+\dfrac12\right)^2-\dfrac14\).

So

\(5-2\left[\left(x+\dfrac12\right)^2-\dfrac14\right]=\dfrac{11}{2}-2\left(x+\dfrac12\right)^2\).

\(5x^2-20x+2=5(x^2-4x)+2\).

\(x^2-4x=(x-2)^2-4\).

So

\(5[(x-2)^2-4]+2=5(x-2)^2-18\).

Write it as

\(4+3x-2x^2=-2\left(x^2-\dfrac32x\right)+4\).

\(x^2-\dfrac32x=\left(x-\dfrac34\right)^2-\dfrac{9}{16}\).

So

\(-2\left[\left(x-\dfrac34\right)^2-\dfrac{9}{16}\right]+4=\dfrac{41}{8}-2\left(x-\dfrac34\right)^2\).

\(3x^2+5x+1=3\left(x^2+\dfrac53x\right)+1\).

\(x^2+\dfrac53x=\left(x+\dfrac56\right)^2-\dfrac{25}{36}\).

So

\(3\left[\left(x+\dfrac56\right)^2-\dfrac{25}{36}\right]+1=3\left(x+\dfrac56\right)^2-\dfrac{13}{12}\).

Divide by 3:

\(x^2+4x+\dfrac23=0\).

\((x+2)^2-\dfrac{10}{3}=0\)

\((x+2)^2=\dfrac{10}{3}\)

\(x+2=\pm \dfrac{\sqrt{30}}{3}\)

\(x=-2\pm \dfrac{\sqrt{30}}{3}\).

Divide by 3:

\(x^2+2x-\dfrac53=0\).

\((x+1)^2-\dfrac83=0\)

\((x+1)^2=\dfrac83\)

\(x+1=\pm \dfrac{2\sqrt6}{3}\)

\(x=-1\pm \dfrac{2\sqrt6}{3}\).

Divide by 5:

\(x^2+10x-\dfrac75=0\).

\((x+5)^2-\dfrac{132}{5}=0\)

\((x+5)^2=\dfrac{132}{5}\)

\(x+5=\pm \dfrac{2\sqrt{165}}{5}\)

\(x=-5\pm \dfrac{2\sqrt{165}}{5}\).

Rearrange:

\(2x^2-20x-3=0\).

Divide by 2:

\(x^2-10x-\dfrac32=0\).

\((x-5)^2-\dfrac{53}{2}=0\)

\((x-5)^2=\dfrac{53}{2}\)

\(x=5\pm \dfrac{\sqrt{106}}{2}\).

Divide by 2:

\(x^2-4x+\dfrac12=0\).

\((x-2)^2-\dfrac72=0\)

\((x-2)^2=\dfrac72\)

\(x=2\pm \dfrac{\sqrt{14}}{2}\).

Divide by 4:

\(x^2-12x-\dfrac{13}{2}=0\).

\((x-6)^2-\dfrac{85}{2}=0\)

\((x-6)^2=\dfrac{85}{2}\)

\(x=6\pm \dfrac{\sqrt{170}}{2}\).

Divide by 2:

\(x^2-\dfrac12x-6=0\).

\(\left(x-\dfrac14\right)^2-\dfrac{97}{16}=0\)

\(\left(x-\dfrac14\right)^2=\dfrac{97}{16}\)

\(x=\dfrac{1\pm \sqrt{97}}{4}\).

Rearrange:

\(3x^2-2x-9=0\).

Divide by 3:

\(x^2-\dfrac23x-3=0\).

\(\left(x-\dfrac13\right)^2-\dfrac{28}{9}=0\)

\(\left(x-\dfrac13\right)^2=\dfrac{28}{9}\)

\(x=\dfrac{1\pm 2\sqrt7}{3}\).