Answer :
To solve the equation [tex]\(\sqrt{b + 20} - \sqrt{b} = 5\)[/tex], we follow these steps:
1. Add [tex]\(\sqrt{b}\)[/tex] to both sides:
[tex]\[ \sqrt{b + 20} = 5 + \sqrt{b} \][/tex]
2. Square both sides to eliminate the radical on the left side:
[tex]\[ (\sqrt{b + 20})^2 = (5 + \sqrt{b})^2 \][/tex]
[tex]\[ b + 20 = 25 + 10\sqrt{b} + b \][/tex]
3. Simplify the equation:
[tex]\[ b + 20 = 25 + 10\sqrt{b} + b \][/tex]
Subtract [tex]\(b\)[/tex] from both sides:
[tex]\[ 20 = 25 + 10\sqrt{b} \][/tex]
4. Isolate the radical term [tex]\(\sqrt{b}\)[/tex]:
[tex]\[ 10\sqrt{b} = 20 - 25 \][/tex]
[tex]\[ 10\sqrt{b} = -5 \][/tex]
Since a square root cannot be negative in real numbers, we need to reevaluate our steps as squaring both sides can introduce extraneous solutions. Therefore, squaring both sides again might be necessary since we likely made an algebraic mistake.
Since the process involves dealing with issues that arise from squaring radicals and verifying the solutions by squaring both sides multiple times to validate the result properly:
- Add [tex]\(\sqrt{b}\)[/tex] to both sides
- Square both sides to remove the radical
- Simplify the equation
- Isolate the radical term
- Square again if needed to clear the issue of extraneous solutions
Thus, the correct statement that describes the necessary process to solve the equation [tex]\(\sqrt{b + 20} - \sqrt{b} = 5\)[/tex]:
Add a constant term to both sides and square both sides twice.
1. Add [tex]\(\sqrt{b}\)[/tex] to both sides:
[tex]\[ \sqrt{b + 20} = 5 + \sqrt{b} \][/tex]
2. Square both sides to eliminate the radical on the left side:
[tex]\[ (\sqrt{b + 20})^2 = (5 + \sqrt{b})^2 \][/tex]
[tex]\[ b + 20 = 25 + 10\sqrt{b} + b \][/tex]
3. Simplify the equation:
[tex]\[ b + 20 = 25 + 10\sqrt{b} + b \][/tex]
Subtract [tex]\(b\)[/tex] from both sides:
[tex]\[ 20 = 25 + 10\sqrt{b} \][/tex]
4. Isolate the radical term [tex]\(\sqrt{b}\)[/tex]:
[tex]\[ 10\sqrt{b} = 20 - 25 \][/tex]
[tex]\[ 10\sqrt{b} = -5 \][/tex]
Since a square root cannot be negative in real numbers, we need to reevaluate our steps as squaring both sides can introduce extraneous solutions. Therefore, squaring both sides again might be necessary since we likely made an algebraic mistake.
Since the process involves dealing with issues that arise from squaring radicals and verifying the solutions by squaring both sides multiple times to validate the result properly:
- Add [tex]\(\sqrt{b}\)[/tex] to both sides
- Square both sides to remove the radical
- Simplify the equation
- Isolate the radical term
- Square again if needed to clear the issue of extraneous solutions
Thus, the correct statement that describes the necessary process to solve the equation [tex]\(\sqrt{b + 20} - \sqrt{b} = 5\)[/tex]:
Add a constant term to both sides and square both sides twice.
