Help with a quadratic equation?

Question

Solve by the square root property. Complete the square first, if necessary.

-4z = 3 - z^2

Answer 1

First, let's arrange our equation in descending powers of z
z^2 -4z-3=0
The factors look like this:
(z.......?)(z.......?)
The ? has to be two numbers that when multiplied together give me -3, and when ADDED together give me -4. (I'm lucky the coefficient of the z^2 term is 1, otherwise it becomes a bit more tricky).
I cannot for the life of me think what they might be!

I shall do what they say: solve by the square root property.

A bit of background first.
a^2 -6a+9=(a-3)^2
x^2 +12x +36 =(x+6)^2
If you do enough of these, you will see a very useful pattern emerge: in a perfect square, the last term is half the coefficient of the middle term squared. This is the secret to completing the square. CAUTION!
The coefficient of the squared term must be +1 for the process to work.

Now back to your problem.
z^2-4z-3 =0
Step1: move the constant term to the other side of the equation, to get z^2-4z =3
Step2: Take half the coefficient of the -4z term and square it. (-4/2)^2=4
Step3: Add this to BOTH SIDES of your equation
Adding to only one side destroys the value of the original equation. z^2-4z+4=3+4
Step4: Tidy up. (z-2)^2=7
Step5: Take the square root of both sides
(z-2)= + or - rt7
Step6: Get +1z all by itself to "solve"
z= 2 +or- rt7
z= 2+rt7 or 2-rt7

I think your teacher is about to show you the general
solution for any quadratic equation. The general form for a quadratic is ax^2 +bx +c =0, and you can always solve any quadratic using the general solution. The formula looks messy, but it really isn't.
My point here is that to get the general solution formula, the expression ax^2+bx+c=0 is solved for x by "completing the square". If you follow the work I've done in my reply to you, you will have no problem following the teacher's development of the general formula. He will
- divide by a to get the coefficient of the x^2 term =1
-move the constant term to the other side
-complete the square

Good luck!

Answer 2

-4z = 3 - z^2

transfer the z^2 onto the other side:
z^2-4z = 3
the perfect square form would be:
(z^2-4z+4)=3
add a 4 to the othr side to make the equation balanced:
(z^2-4z+4)=3+4
factor and simplify:

(z-2)^2=7

Answer 3

-4z = 3 - z^2

z² - 4z - 3 = 0

Complete the square

z² - 4z - 3 + 7 = 7

z² - 4z + 4 = 7

(z - 2)² = 7

Take the square root of both sides

(z - 2) = ±√7

z = 2 ±√7
.

Answer 4

rearrange the terms. Move the terms on the right side to the left and change their signs. You get z^2-4z-3=0

Then, use the quadratic formula where a=1, b=-4, c=-3.