Sure! Let's work through a quadratic equation involving the variable \( z \).
Consider the quadratic equation:
\[ 2z^2 - 4z - 6 = 0 \]
Step 1: Identify coefficients
In the standard form of a quadratic equation \( az^2 + bz + c = 0 \), we have:
- \( a = 2 \)
- \( b = -4 \)
- \( c = -6 \)
Step 2: Use the quadratic formula
The solutions for \( z \) can be found using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Step 3: Calculate the discriminant
First, we calculate the discriminant \( D = b^2 - 4ac \):
\[ D = (-4)^2 - 4 \cdot 2 \cdot (-6) \] \[ D = 16 + 48 = 64 \]
Step 4: Substitute values into the formula
Now, substituting the values back into the quadratic formula:
\[ z = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 2} \] \[ z = \frac{4 \pm 8}{4} \]
Step 5: Solve for the two values of \( z \)
Now, we calculate the two potential solutions:
- \( z_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3 \)
- \( z_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1 \)
Step 6: Final solutions
Thus, the solutions for the equation \( 2z^2 - 4z - 6 = 0 \) are:
\[ z = 3 \quad \text{and} \quad z = -1 \]
If you have a different equation in mind or need further assistance, feel free to ask!