Question
o
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.
remove frac pm sqrt / } { \
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.
remove frac pm sqrt / } { \
Answers
GPT-4o mini
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.
\[
\frac{b - 5}{-2b + 5} = \frac{50}{b^2 - 25},
\]
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.