Thanks for the correction. The correct equation is:

\[
(t - 3)^2 + 2t - 15 = 2
\]

Let's solve this step by step.

### Step 1: Expand the squared term

First, expand \((t - 3)^2\):

\[
(t - 3)^2 = t^2 - 6t + 9
\]

### Step 2: Substitute the expanded form into the equation

Substitute this into the original equation:

\[
t^2 - 6t + 9 + 2t - 15 = 2
\]

### Step 3: Simplify the equation

Combine like terms:

\[
t^2 - 4t - 6 = 2
\]

### Step 4: Move all terms to one side

Subtract 2 from both sides to set the equation to zero:

\[
t^2 - 4t - 8 = 0
\]

### Step 5: Solve using the quadratic formula

Now, use the quadratic formula to solve for \(t\):

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(t^2 - 4t - 8 = 0\):

- \(a = 1\)
- \(b = -4\)
- \(c = -8\)

### Step 6: Calculate the discriminant and the roots

Calculate the discriminant:

\[
\Delta = b^2 - 4ac = (-4)^2 - 4(1)(-8) = 16 + 32 = 48
\]

Since the discriminant is positive, the equation has two real roots. The square root of 48 can be simplified:

\[
\sqrt{48} = 4\sqrt{3}
\]

So the roots are:

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

### Final Answer

The solutions to the equation are:

\[
t = 2 + 2\sqrt{3} \quad \text{or} \quad t = 2 - 2\sqrt{3}
\]

Would you like more details or have any questions?

---

Here are five related questions you might find interesting:

1. How do you simplify expressions involving square roots like \(\sqrt{48}\)?
2. What is the significance of the discriminant when it is not a perfect square?
3. How would you solve this equation by completing the square instead of using the quadratic formula?
4. Can this equation be factored, or is the quadratic formula the best approach?
5. What are the applications of solving quadratic equations in real-life problems?

**Tip:** When solving quadratic equations, always check whether the discriminant is a perfect square to decide if the equation can be factored easily.

Learn how to solve the quadratic equation (t-3)^2 + 2t - 15 = 2 step-by-step with a detailed solution. This problem covers key algebraic concepts, including quadratic equations, completing the square, and the quadratic formula. Suitable for high school students.

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