Radical Equations

1. Isolate the radical: $$ \sqrt{x + 3} = 5 $$ (already isolated).

2. Eliminate the radical: Square both sides: $$ (\sqrt{x + 3})^2 = 5^2 $$.

$$ x + 3 = 25 $$

3. Solve for $$ x $$:

$$ x = 25 - 3 = 22 $$

4. Check the solution by substituting $$ x = 22 $$ back into the original equation:

$$ \sqrt{22 + 3} = \sqrt{25} = 5 $$

Since both sides are equal, $$ x = 22 $$ is a valid solution.

Steps to Solve:

1. Eliminate the Radicals: Square both sides to remove both radicals:
   $$ (\sqrt{3x + 1})^2 = (\sqrt{x + 7})^2 $$
   $$ 3x + 1 = x + 7 $$
2. Solve the Resulting Equation: Solve for \(x\):
   $$ 2x = 6 $$
   $$ x = 3 $$
3. Check for Extraneous Solutions: Substitute \(x = 3\) back into the original equation to verify it is a valid solution.

1. Isolate the radical: $$ \sqrt[3]{2x - 1} = 3 $$ (already isolated).

2. Eliminate the radical: Cube both sides: $$ (\sqrt[3]{2x - 1})^3 = 3^3 $$.

$$ 2x - 1 = 27 $$

3. Solve for $$ x $$:

$$ 2x = 27 + 1 = 28 $$

$$ x = \frac{28}{2} = 14 $$

4. Check the solution by substituting $$ x = 14 $$ back into the original equation:

$$ \sqrt[3]{2(14) - 1} = \sqrt[3]{28 - 1} = \sqrt[3]{27} = 3 $$

Since both sides are equal, $$ x = 14 $$ is a valid solution.

Steps to Solve:

1. Isolate the Radical: The radical is already isolated on one side.
2. Eliminate the Radical: Square both sides to remove the square root:
   $$ (\sqrt{2x + 7})^2 = (3x - 5)^2 $$
   $$ 2x + 7 = 9x^2 - 30x + 25 $$
3. Solve the Resulting Quadratic Equation: Bring all terms to one side to form a quadratic equation:
   $$ 0 = 9x^2 - 32x + 18 $$
4. Check for Extraneous Solutions: Solve the quadratic equation and check the roots in the original equation to ensure they are valid solutions.

Steps to Solve:

1. Isolate One of the Radicals: Move one radical to the other side:
   $$ \sqrt{x + 3} = 4 - \sqrt{x - 2} $$
2. Eliminate the Radical: Square both sides to remove the first radical:
   $$ (\sqrt{x + 3})^2 = (4 - \sqrt{x - 2})^2 $$
   $$ x + 3 = 16 - 8\sqrt{x - 2} + (x - 2) $$
3. Isolate the Remaining Radical: Combine like terms and isolate the radical:
   $$ 8\sqrt{x - 2} = 21 - 2x $$
4. Eliminate the Radical Again: Square both sides again to eliminate the second radical:
   $$ 64(x - 2) = (21 - 2x)^2 $$
5. Solve the Resulting Equation: Simplify and solve the resulting equation. Finally, check the solutions to ensure they are valid.

Steps to Solve:

1. Isolate the Radical: The radical is already isolated on one side.
2. Eliminate the Radical: Square both sides to remove the square root:
   $$ (\sqrt{x^2 + 4x + 4})^2 = (x + 6)^2 $$
   $$ x^2 + 4x + 4 = x^2 + 12x + 36 $$
3. Solve the Resulting Equation: Simplify and solve the linear equation:
   $$ 4x + 4 = 12x + 36 $$
   $$ -8x = 32 $$
   $$ x = -4 $$
4. Check for Extraneous Solutions: Substitute \(x = -4\) back into the original equation to verify it is a valid solution.

Steps to Solve:

1. Isolate the Radical: The radical is already isolated on one side.
2. Eliminate the Radical: Cube both sides to remove the cube root:
   $$ (\sqrt[3]{2x^2 - 3x + 7})^3 = (x + 2)^3 $$
   $$ 2x^2 - 3x + 7 = x^3 + 6x^2 + 12x + 8 $$
3. Solve the Resulting Equation: Rearrange the terms to form a cubic equation:
   $$ x^3 + 4x^2 + 15x + 1 = 0 $$
4. Solve the Cubic Equation: Use methods such as factoring, synthetic division, or numerical approaches to solve for \(x\).
5. Check for Extraneous Solutions: Ensure that all solutions satisfy the original equation.

Steps to Solve:

1. Isolate the Radical: The radical is already isolated on one side.
2. Eliminate the Radical: Square both sides to remove the square root:
   $$ \frac{2x + 3}{x - 1} = 16 $$
3. Solve the Resulting Rational Equation: Multiply both sides by \(x - 1\):
   $$ 2x + 3 = 16(x - 1) $$
   $$ 2x + 3 = 16x - 16 $$
4. Simplify and Solve for \(x\):
   $$ 14x = 19 $$
   $$ x = \frac{19}{14} $$
5. Check for Extraneous Solutions: Ensure the solution is valid by substituting it back into the original equation.