To solve the equation \( 4d^2 + 5d - 9 = 0 \), we can use the quadratic formula:
\[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 4 \), \( b = 5 \), and \( c = -9 \).
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = 5^2 = 25 \] \[ 4ac = 4 \times 4 \times (-9) = -144 \] \[ b^2 - 4ac = 25 - (-144) = 25 + 144 = 169 \]
Now, we can substitute back into the quadratic formula:
\[ d = \frac{-5 \pm \sqrt{169}}{2 \times 4} \] \[ d = \frac{-5 \pm 13}{8} \]
This gives us two possible solutions:
- \( d = \frac{-5 + 13}{8} = \frac{8}{8} = 1 \)
- \( d = \frac{-5 - 13}{8} = \frac{-18}{8} = -\frac{9}{4} \)
So the solutions for \( d \) are:
\[ d = 1 \quad \text{and} \quad d = -\frac{9}{4} \]