We are tasked with solving the quadratic equation \( h(t) = -t^2 + 6t - 7 \), which is a standard quadratic equation. The general form of a quadratic equation is \( at^2 + bt + c = 0 \), where \( a \), \( b \), and \( c \) are constants. 

For our equation:
- \( a = -1 \)
- \( b = 6 \)
- \( c = -7 \)

We'll use the quadratic formula to find the solutions for \( t \). The quadratic formula is given by:

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

### Step 1: Identify the discriminant
The discriminant is the part inside the square root: \( \Delta = b^2 - 4ac \). Substituting the values for \( a \), \( b \), and \( c \):

\[
\Delta = 6^2 - 4(-1)(-7) = 36 - 28 = 8
\]

### Step 2: Apply the quadratic formula
Now that we know the discriminant, we can substitute it into the quadratic formula:

\[
t = \frac{-6 \pm \sqrt{8}}{2(-1)}
\]

Simplifying:

\[
t = \frac{-6 \pm 2\sqrt{2}}{-2}
\]

Now divide both terms by \(-2\):

\[
t = 3 \mp \sqrt{2}
\]

### Final solution:
The solutions in radical form are:

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

Let me know if you would like a deeper explanation or further details.

Here are five related questions to deepen your understanding:
1. How does the discriminant affect the type of solutions in a quadratic equation?
2. What happens if the discriminant is negative?
3. Can the quadratic formula be used for any quadratic equation?
4. How do you simplify expressions involving square roots like \( \sqrt{8} \)?
5. How would the solution change if the equation were \( t^2 - 6t + 7 = 0 \)?

**Tip**: Always check the sign of the coefficient \( a \) in quadratic equations, as it affects the concavity of the graph.

solve h(t) = -t^2 + 6t - 7 in radical form

Learn how to solve the quadratic equation h(t) = -t^2 + 6t - 7 in radical form using the quadratic formula. This detailed step-by-step solution covers key algebraic concepts such as finding the discriminant and simplifying radicals. Suitable for students in Grades 8-10.

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