the following step numbers are circled in the worksheet.
1.
your original equation
2.
subtract 1 from both sides of the equation to isolate the constant term.
3.
factor out the coefficient of the x^2 term so that the coefficient of the x^2 term in the expression that is to be factored is equal to 1. this is a necessary step in completing the square on the expression inside the parentheses.
4.
complete the square on the expression inside the parentheses.
see below the worksheet for how this step is performed and why.
5.
simplify the expression by distributing the multiplication on the left side of the equation.
6.
add 8 to both dies of the equation to get all the constant terms to the right side of the equation.
7.
divide both sides of the equation by 2 to isolate the squared term to the left side of the equation.
8.
take the square root of both sides of the equation.
don't forget that you need plus or minus the square root on the right side of the equation.
9.
subtract 2 from both sides of the equation to solve for x.
the results shown are in fraction format and decimal format.
the following steps are performed in step number 4 in the worksheet.
start with 2 * (x^2 + 4x) = -1.
the expression to complete the square on is x^2 + 4x.
you take the square root of the x^2 term and half the coefficient of the x term and then square them.
you will get x^2 + 4x = (x+2)^2
you then subtract the square of the constant term and subtract it.
you will get x^2 + 4x = (x+2)^2 - 4.
why is this done?
here's why:
(x+2)^2 = x^2 + 4x + 4.
but you want (x+2)^2 to be equal to x^2 + 4x.
you have an extra 4 that needs to be subtracted out.
that's why you get x^2 + 4x = (x+2)^2 - 4.
(x+2)^2 - 4 = x^2 + 4x + 4 - 4 which results in x^2 + 4x.
so, once again.
start with 2 * (x^2 + 4x) = -1.
the expression to complete the square on is x^2 + 4x.
take the square root of the x^2 term and half the coefficient of the x term to get:
x^2 + 4x = (x+2)^2
subtract the square of the constant term of this result to get:
x^2 + 4x = (x+2)^2 - 4
return to your original equation of 2 * (x^2 + 4x) = -1 and replace x^2 + 4x with (x+2)^2 - 4 to get:
2 * (x^2 + 4x) = -1 becomes 2 * ( (x+2)^2 - 4) ) = -1
distribute the multiplication to get 2 * (x+2)^2 - 2 * 4 = -1
simplify this to get 2 * (x+2)^2 - 8 = -1
this is the result shown in step 5.
you can also graph your equation.
you will get the same answer.
that graph is shown below:
you can also look at the following reference to see how this is done from another source.
http://www.purplemath.com/modules/sqrquad.htm
there are lots more references on the web.
just do a search on "completing the square method".