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Question

To solve the equation \( \frac{b - 5}{-2b + 5} = \frac{50}{b^2 - 25} \), let's start by rewriting the equation:
\[
\frac{b - 5}{-2b + 5} = \frac{50}{b^2 - 25}
\]

### Step 1: Factor the Denominator

We notice that \( b^2 - 25 \) can be factored since it's a difference of squares:

\[
b^2 - 25 = (b - 5)(b + 5)
\]

Now 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 becomes \( (b - 5)^2 \):

\[
(b - 5)^2 = b^2 - 10b + 25
\]

The right side expands as follows:

\[
50(-2b + 5) = -100b + 250
\]

### Step 4: Set the equation

Now we set the two expressions equal to each other:

\[
b^2 - 10b + 25 = -100b + 250
\]

### Step 5: Move all terms to one side

Now, combine like terms by moving everything to one side of the equation:

\[
b^2 - 10b + 25 + 100b - 250 = 0
\]

This simplifies to:

\[
b^2 + 90b - 225 = 0
\]

### Step 6: Solve the quadratic equation

Now, we can apply the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where:

- \( A = 1 \)
- \( B = 90 \)
- \( C = -225 \)

Let's calculate the discriminant:

\[
B^2 - 4AC = 90^2 - 4(1)(-225) = 8100 + 900 = 9000
\]

Now, apply the quadratic formula:

\[
b = \frac{-90 \pm \sqrt{9000}}{2}
\]

The square root of \( 9000 \) can be simplified:

\[
\sqrt{9000} = \sqrt{90^2 \times 10} = 90 \sqrt{10}
\]

So, we have:

\[
b = \frac{-90 \pm 90 \sqrt{10}}{2} = -45 \pm 45 \sqrt{10}
\]

### Step 7: Final Result

Therefore, the solutions for \( b \) are:

\[
b = -45 + 45\sqrt{10} \, \text{and} \, b = -45 - 45\sqrt{10}
\]

This completes the solution of the equation.
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Answer 1

To solve the equation

\[ \frac{b - 5}{-2b + 5} = \frac{50}{b^2 - 25}, \]

we start by noting that the denominator on the right-hand side can be factored:

\[ b^2 - 25 = (b - 5)(b + 5). \]

We can rewrite the equation as:

\[ \frac{b - 5}{-2b + 5} = \frac{50}{(b - 5)(b + 5)}. \]

Next, we cross-multiply to eliminate the fractions:

\[ (b - 5)(b - 5) = 50(-2b + 5). \]

Now we expand both sides. The left side is:

\[ (b - 5)^2 = b^2 - 10b + 25. \]

The right side simplifies to:

\[ 50(-2b + 5) = -100b + 250. \]

Setting the two sides equal gives us:

\[ b^2 - 10b + 25 = -100b + 250. \]

Now, we will move all terms to one side of the equation:

\[ b^2 - 10b + 25 + 100b - 250 = 0. \]

This simplifies to:

\[ b^2 + 90b - 225 = 0. \]

Now, we can use the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 1 \), \( B = 90 \), and \( C = -225 \).

Calculating the discriminant:

\[ B^2 - 4AC = 90^2 - 4(1)(-225) = 8100 + 900 = 9000. \]

Now applying the quadratic formula:

\[ b = \frac{-90 \pm \sqrt{9000}}{2}. \]

The square root of \( 9000 \) can be simplified:

\[ \sqrt{9000} = \sqrt{90^2 \times 10} = 90 \sqrt{10}. \]

Thus, we have:

\[ b = \frac{-90 \pm 90 \sqrt{10}}{2} = -45 \pm 45 \sqrt{10}. \]

Therefore, the final solutions for \( b \) are:

\[ b = -45 + 45\sqrt{10} \quad \text{and} \quad b = -45 - 45\sqrt{10}. \]

These are the solutions to the equation.