Question

Solve using zero product property
N^2 + 13n +40=0

Answer 1

To solve the quadratic equation \( N^2 + 13N + 40 = 0 \) using the zero product property, we first need to factor the quadratic expression.

We are looking for two numbers that multiply to \( 40 \) (the constant term) and add up to \( 13 \) (the coefficient of \( N \)).

The numbers that meet these criteria are \( 8 \) and \( 5 \) because:

- \( 8 \times 5 = 40 \)
- \( 8 + 5 = 13 \)

Now, we can factor the quadratic:

\[
N^2 + 13N + 40 = (N + 8)(N + 5)
\]

Setting the factored expression to zero gives us:

\[
(N + 8)(N + 5) = 0
\]

Using the zero product property, we set each factor equal to zero:

1. \( N + 8 = 0 \)

\[
N = -8
\]

2. \( N + 5 = 0 \)

\[
N = -5
\]

Thus, the solutions to the equation \( N^2 + 13N + 40 = 0 \) are:

\[
N = -8 \quad \text{and} \quad N = -5
\]