Question

Solve 5d
2
+3d=8 by factoring and using the zero product property.

d=−1.6, 1

d=1.6, −1

d=0, 3

d=0, −3

Answer 1

To solve the equation \( 5d^2 + 3d = 8 \) by factoring and using the zero product property, we first rearrange the equation to set it to zero:

\[
5d^2 + 3d - 8 = 0
\]

Next, we need to factor the quadratic equation. We look for two numbers that multiply to \( 5 \times (-8) = -40 \) and add to \( 3 \). The numbers that satisfy this condition are \( 8 \) and \( -5 \) since:

\[
8 \cdot (-5) = -40 \quad \text{and} \quad 8 + (-5) = 3
\]

Now we can rewrite the middle term \( 3d \) using \( 8d \) and \( -5d \):

\[
5d^2 + 8d - 5d - 8 = 0
\]

Next, we group the terms:

\[
(5d^2 + 8d) + (-5d - 8) = 0
\]

Factoring by grouping, we have:

\[
d(5d + 8) - 1(5d + 8) = 0
\]

Factoring out \( (5d + 8) \):

\[
(5d + 8)(d - 1) = 0
\]

Now we can apply the zero product property. This gives us two equations to solve:

1. \( 5d + 8 = 0 \)
2. \( d - 1 = 0 \)

Solving the first equation:

\[
5d + 8 = 0 \implies 5d = -8 \implies d = -\frac{8}{5} \implies d = -1.6
\]

Solving the second equation:

\[
d - 1 = 0 \implies d = 1
\]

Thus, the solutions to the equation \( 5d^2 + 3d - 8 = 0 \) are:

\[
d = -1.6 \quad \text{and} \quad d = 1
\]

The correct answer is:

**d=−1.6, 1**