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:
- \( 5d + 8 = 0 \)
- \( d - 1 = 0 \)
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