Solve the equation for all real solutions in simplest form.

3, z, squared, minus, z, plus, 5, equals, 6

3*z*2−*z*+5=

6

To solve the equation:

\[
3z^2 - z + 5 = 6
\]

### Step 1: Move all terms to one side

Subtract 6 from both sides to set the equation to zero:

\[
3z^2 - z + 5 - 6 = 0
\]

Simplify:

\[
3z^2 - z - 1 = 0
\]

### Step 2: Solve the quadratic equation

We now have a standard quadratic equation:

\[
3z^2 - z - 1 = 0
\]

To solve this, we use the quadratic formula:

\[
z = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3}
\]

Simplify step-by-step:

1. Calculate the discriminant:
   \[
   (-1)^2 - 4 \cdot 3 \cdot (-1) = 1 + 12 = 13
   \]

2. Substitute into the formula:
   \[
   z = \frac{1 \pm \sqrt{13}}{6}
   \]

### Final Answer

The solutions are:

\[
z = \frac{1 + \sqrt{13}}{6} \quad \text{and} \quad z = \frac{1 - \sqrt{13}}{6}
\]

These are the real solutions in simplest form.

Would you like further details or have any questions?

Here are 5 questions to further your understanding:
1. How can we check if these solutions are correct?
2. What does the discriminant tell us about the number of real solutions?
3. Why is the quadratic formula useful for solving equations in standard form?
4. Can the solutions be simplified further, or are they in simplest form?
5. What would happen if the discriminant were negative?

**Tip**: Always calculate the discriminant first when using the quadratic formula to see if real solutions exist!

Solve the equation for all real solutions in simplest form. 3z^2 - z + 5 = 6

Master solving the quadratic equation 3z^2 - z + 5 = 6 with a step-by-step guide. This problem explores algebraic concepts such as the quadratic formula and discriminant analysis, suitable for students in Grades 9-10.

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