To solve the equation \( \frac{b - 5}{-2b + 5} = \frac{50}{b^2 - 25} \), let's start by rewriting the equation properly:
\[ \frac{b - 5}{-2b + 5} = \frac{50}{b^2 - 25} \]
Step 1: Factor the Denominator
We notice that \( b^2 - 25 \) can be factored, as it is a difference of squares:
\[ b^2 - 25 = (b - 5)(b + 5) \]
Now we can re-write our equation:
\[ \frac{b - 5}{-2b + 5} = \frac{50}{(b - 5)(b + 5)} \]
Step 2: Cross-multiply
Next, we can cross-multiply to eliminate the fractions:
\[ (b - 5)(b - 5) = 50(-2b + 5) \]
Step 3: Expand both sides
The left side simplifies to \( (b - 5)^2 \):
\[ (b - 5)^2 = b^2 - 10b + 25 \]
The right side expands as follows:
\[ 50(-2b + 5) = -100b + 250 \]
Step 4: Set the equation
Now we set the two sides equal to each other:
\[ b^2 - 10b + 25 = -100b + 250 \]
Step 5: Move all terms to one side
Bringing all terms to one side gives:
\[ b^2 - 10b + 25 + 100b - 250 = 0 \]
This simplifies to:
\[ b^2 + 90b - 225 = 0 \]
Step 6: Solve the quadratic equation
Next, we use the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where:
- \( A = 1 \)
- \( B = 90 \)
- \( C = -225 \)
Calculating the discriminant:
\[ B^2 - 4AC = 90^2 - 4(1)(-225) = 8100 + 900 = 9000 \]
Now we apply the quadratic formula:
\[ b = \frac{-90 \pm \sqrt{9000}}{2} \]
The square root of \( 9000 \) can be simplified:
\[ \sqrt{9000} = \sqrt{90^2 \cdot 10} = 90\sqrt{10} \]
Therefore, substituting back into the formula gives us:
\[ b = \frac{-90 \pm 90\sqrt{10}}{2} = -45 \pm 45\sqrt{10} \]
Step 7: Final Result
Thus, the solutions for \( b \) are:
\[ b = -45 + 45\sqrt{10} \quad \text{and} \quad b = -45 - 45\sqrt{10} \]
These represent the complete solution to the equation.
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