z 2 −5z−5=3z

Sure, let's solve the equation step by step:

1. First, let's rearrange the equation to bring all terms to one side. This gives us:

   z^2 - 5z - 3z - 5 = 0

2. Simplify the equation by combining like terms:

   z^2 - 8z - 5 = 0

3. Now, we need to solve for z. This is a quadratic equation, so we can use the quadratic formula, which is:

   z = [-b ± sqrt(b^2 - 4ac)] / (2a)

   In this equation, a = 1 (the coefficient of z^2), b = -8 (the coefficient of z), and c = -5.

4. Plug these values into the quadratic formula:

   z = [8 ± sqrt((-8)^2 - 41(-5))] / (2*1) z = [8 ± sqrt(64 + 20)] / 2 z = [8 ± sqrt(84)] / 2

5. Simplify the square root:

   z = [8 ± 2*sqrt(21)] / 2

6. Simplify the equation by dividing everything inside the brackets by 2:

   z = 4 ± sqrt(21)

So, the solutions to the equation z^2 - 8z - 5 = 0 are z = 4 + sqrt(21) and z = 4 - sqrt(21).