To solve the quadratic equation \( y^2 - 3y = 9 \), let's follow these steps:

### Step 1: Rearrange the equation
First, bring all terms to one side of the equation to set it equal to zero:
\[
y^2 - 3y - 9 = 0
\]

### Step 2: Solve using the quadratic formula
The quadratic formula is given by:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( y^2 - 3y - 9 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -3 \)
- \( c = -9 \)

Now, substitute these values into the quadratic formula:

\[
y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-9)}}{2(1)}
\]

Simplify inside the square root:

\[
y = \frac{3 \pm \sqrt{9 + 36}}{2}
\]
\[
y = \frac{3 \pm \sqrt{45}}{2}
\]

### Step 3: Simplify the square root
\[
\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]

So, the equation becomes:
\[
y = \frac{3 \pm 3\sqrt{5}}{2}
\]

### Step 4: Express the two solutions
The two possible values for \( y \) are:
\[
y_1 = \frac{3 + 3\sqrt{5}}{2}, \quad y_2 = \frac{3 - 3\sqrt{5}}{2}
\]

Thus, the solutions to the equation \( y^2 - 3y = 9 \) are:
\[
y = \frac{3 + 3\sqrt{5}}{2} \quad \text{or} \quad y = \frac{3 - 3\sqrt{5}}{2}
\]

Would you like more details or have any questions?

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Here are 5 related questions to extend the topic:
1. How do you complete the square for a quadratic equation like \( y^2 - 3y - 9 = 0 \)?
2. Can you graph the quadratic function \( y^2 - 3y - 9 \)?
3. How does the discriminant help determine the nature of the roots of a quadratic equation?
4. What is the significance of the vertex in the graph of a quadratic function?
5. How would you solve \( y^2 - 3y = 9 \) by factoring instead of using the quadratic formula?

**Tip**: Always check if a quadratic equation can be factored easily before applying the quadratic formula!

This detailed guide explains how to solve the quadratic equation y^2 - 3y = 9. Learn step-by-step how to rearrange the equation, apply the quadratic formula, and simplify the solution. Suitable for students in Grades 9-10.

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