Completing the square is pretty hard to explain for me but I'll try my best and hopefully you get what I'm saying in these steps.
First make sure your numbers are all on one side leaving zero on the other side:
z² - 12z + 11 = 0
The next step is the tricky part to explain. So look on the first number and the third number in the formula. The first number is 1z². What multiples equal 1? The only multiples would be one times one (1 x 1). So below z² write one over one like this:
z² - 12z + 11 = 0
[tex] \frac{1}{1} [/tex]
But remember this is 1 x 1 and not actually division. For the + 11, think of all the multiples that equal positive 11. It can be 1 x 11 or -1 x -11 so put the first option in fraction form first to test. It doesn't matter which is on top or bottom.
So now you should be left with these:
[tex] \frac{1}{1} \frac{11}{1}[/tex]
Now you need to cross multiply here. 1 x 11 and 1 x 1. That will give you two positive numbers 1 and 11. In order to get your middle number from step 1 you need to add these two numbers and see if you will get -12. As you can see 11 + 1 doesn't equal -12 so now we need to use our second option -1 x -11. Our paper should now be changed to this:
[tex] \frac{1}{1} \frac{-11}{-1}[/tex]
Cross multiply. 1 x -11 and 1 x -1. You get -11 and -1 as the result. Add both together and you get -12.
Now you just need to rearrange your fractions here. you can replace your 1 over one to z over z since z x z still equals your problem's z²:
[tex] \frac{1}{1} \frac{-11}{-1} \\ \\ \frac{z}{z} \frac{-11}{-1} \\ \\ (z-11)(z-1)=0[/tex]
Finally solve for z. Separate parentheses to two separate equations like this:
z - 11 = 0 and z - 1 = 0
z = 11 and z = 1
That should be your answer. Sorry if it still doesn't make sense still.
