Solve each equation. √(6x+7) - 9 = x-7

Verified step by step guidance

1. Start by isolating the square root expression on one side of the equation. Add 9 to both sides to get: \(\sqrt{6x + 7} = x - 7 + 9\).
2. Simplify the right side of the equation: \(\sqrt{6x + 7} = x + 2\).
3. To eliminate the square root, square both sides of the equation: \((\sqrt{6x + 7})^2 = (x + 2)^2\).
4. Simplify both sides: \$6x + 7 = (x + 2)(x + 2) = x^2 + 4x + 4$.
5. Rearrange the equation to set it equal to zero: \$0 = x^2 + 4x + 4 - 6x - 7$, then combine like terms to form a quadratic equation.

Key Concepts

Solving Radical Equations

Radical equations involve variables inside a root, often a square root. To solve them, isolate the radical expression on one side before squaring both sides to eliminate the root. This process may introduce extraneous solutions, so checking all solutions in the original equation is essential.

Isolating Variables

Isolating the variable means manipulating the equation to get the variable alone on one side. This step is crucial before squaring both sides in radical equations to avoid complicating the expression and to simplify solving for the variable.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that do not satisfy the original equation. After finding potential solutions, substitute them back into the original equation to verify which are valid and discard any extraneous ones.