To solve the quadratic equation:

\[
t^2 + 9t + 14 = 0
\]

we can use factoring. We need to find two numbers that multiply to 14 and add up to 9.

The factors of 14 that add up to 9 are 7 and 2. Thus, we can rewrite the equation as:

\[
(t + 7)(t + 2) = 0
\]

Now, set each factor equal to zero:

1. \( t + 7 = 0 \Rightarrow t = -7 \)
2. \( t + 2 = 0 \Rightarrow t = -2 \)

So, the solution set is:

\[
t = -7 \quad \text{and} \quad t = -2
\]

Would you like further details, or do you have any questions?

Here are five related questions you might find interesting:

1. What if we used the quadratic formula to solve this equation?
2. How would the solution change if the equation were \( t^2 + 9t - 14 = 0 \)?
3. Can we solve for complex roots if the factors don't exist in real numbers?
4. How can we verify the solutions by substituting back into the original equation?
5. What is the significance of the vertex in a quadratic equation like this?

**Tip:** When factoring, always look for pairs of numbers that multiply to the constant term and add up to the linear term's coefficient.

Learn how to solve the quadratic equation t^2 + 9t + 14 = 0 using factoring. This solution involves finding factors that multiply to 14 and add to 9, leading to the solution set t = -7 and t = -2. Suitable for students in Grades 7-9.

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