Solve the equation for all real solutions in simplest form.minus, 3, z, squared, plus, 4, z, minus, 2, equals, minus, 6, z, squared−3z 2 +4z−2=−6z 2

The given equation is -3z^2 + 4z - 2 = -6z^2.

Step 1: Add 6z^2 to both sides of the equation to get rid of the negative sign on the right side of the equation. This gives us:

-3z^2 + 6z^2 + 4z - 2 = 0

Step 2: Simplify the equation:

3z^2 + 4z - 2 = 0

Step 3: Now, we can solve this quadratic equation by using the quadratic formula, which is given by:

z = [-b ± sqrt(b^2 - 4ac)] / 2a

where a = 3, b = 4, and c = -2.

Step 4: Substitute the values of a, b, and c into the quadratic formula:

z = [-4 ± sqrt((4)^2 - 43(-2))] / 2*3 z = [-4 ± sqrt(16 + 24)] / 6 z = [-4 ± sqrt(40)] / 6 z = [-4 ± 2sqrt(10)] / 6

Step 5: Simplify the equation to get the solutions:

z = -2/3 ± sqrt(10)/3

So, the solutions to the equation are z = -2/3 + sqrt(10)/3 and z = -2/3 - sqrt(10)/3.