Evaluate the expression \(\frac{d^2 - 5d + 25}{2d^3 + 250}\) for \(d = -4.5\).

Solution:

The expression is \( \frac{d^2 - 5d + 25}{2d^3 + 250} \).

Factor out 2 from the denominator: \(2(d^3 + 125)\).

Factor the sum of cubes in the parenthesis: \(d^3 + 5^3 = (d+5)(d^2 - 5d + 25)\).

So, the denominator is \(2(d+5)(d^2 - 5d + 25)\).

The expression becomes \( \frac{d^2 - 5d + 25}{2(d+5)(d^2 - 5d + 25)} \).

For \(d^2 - 5d + 25
eq 0\) (the discriminant is \((-5)^2 - 4(1)(25) = 25 - 100 = -75 < 0\), so this term is never zero), we can cancel out \(d^2 - 5d + 25\), simplifying the expression to \( \frac{1}{2(d+5)} \).

Now, substitute \(d = -4.5\):

\[ \frac{1}{2(-4.5 + 5)} = \frac{1}{2(0.5)} = \frac{1}{1} = 1 \]

Answer: 1