To solve the radical equation \( \sqrt{b + 14} = \sqrt{5b - 14} \), we will start by squaring both sides of the equation to eliminate the square roots.
\[ \left( \sqrt{b + 14} \right)^2 = \left( \sqrt{5b - 14} \right)^2 \]
This simplifies to:
\[ b + 14 = 5b - 14 \]
Next, we'll rearrange the equation to isolate \( b \):
\[ 14 + 14 = 5b - b \]
Which simplifies to:
\[ 28 = 4b \]
Now, divide both sides by 4 to solve for \( b \):
\[ b = \frac{28}{4} = 7 \]
Now that we have a solution, we need to check for any extraneous solutions by substituting \( b = 7 \) back into the original equation:
Original equation:
\[ \sqrt{b + 14} = \sqrt{5b - 14} \]
Substituting \( b = 7 \):
Left side:
\[ \sqrt{7 + 14} = \sqrt{21} \]
Right side:
\[ \sqrt{5(7) - 14} = \sqrt{35 - 14} = \sqrt{21} \]
Both sides are equal, which confirms that \( b = 7 \) is a valid solution.
Thus, the solution to the radical equation is:
\[ \boxed{7} \]
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