To solve the equation \( b^2 - 5 - 2b + 5 = 50b^2 - 25 \), we start by simplifying both sides.
The left-hand side can be simplified as follows:
- Combine like terms: \( b^2 - 2b - 5 + 5 = b^2 - 2b \).
So, the equation becomes: \[ b^2 - 2b = 50b^2 - 25. \]
Next, we move all terms to one side of the equation to set it to zero:
- Subtract \( 50b^2 \) and add \( 25 \) to both sides: \[ b^2 - 2b - 50b^2 + 25 = 0. \]
Now combine like terms: \[ -49b^2 - 2b + 25 = 0. \]
To make calculations easier, we can multiply the entire equation by -1 to change the signs: \[ 49b^2 + 2b - 25 = 0. \]
Next, we can use the quadratic formula to find the roots of this equation. The quadratic formula is: \[ b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}, \] where \( A = 49 \), \( B = 2 \), and \( C = -25 \).
- Calculate the discriminant: \[ B^2 - 4AC = 2^2 - 4(49)(-25) = 4 + 4900 = 4904. \]
Now we find \( b \) using the quadratic formula:
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Substitute into the formula: \[ b = \frac{-2 \pm \sqrt{4904}}{2(49)}. \]
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Simplify \( \sqrt{4904} \): Since \( 4904 = 4 \cdot 1226 \), we have \( \sqrt{4904} = 2\sqrt{1226} \).
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Substitute back: \[ b = \frac{-2 \pm 2\sqrt{1226}}{98} = \frac{-1 \pm \sqrt{1226}}{49}. \]
Thus, the solutions for \( b \) are: \[ b = \frac{-1 + \sqrt{1226}}{49} \quad \text{and} \quad b = \frac{-1 - \sqrt{1226}}{49}. \]
These are the values of \( b \) that satisfy the original equation.
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