Solve each equation. √(3x+7) = 3x+5

Verified step by step guidance

1. Start with the given equation: \(\sqrt{3x + 7} = 3x + 5\).
2. To eliminate the square root, square both sides of the equation: \(\left(\sqrt{3x + 7}\right)^2 = (3x + 5)^2\).
3. Simplify both sides: \$3x + 7 = (3x + 5)^2$.
4. Expand the right side using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \$3x + 7 = 9x^2 + 30x + 25$.
5. Rearrange the equation to set it equal to zero: \$0 = 9x^2 + 30x + 25 - 3x - 7$, then combine like terms to get 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 and then eliminate the root by raising both sides to the appropriate power, typically squaring for square roots. This process can introduce extraneous solutions, so checking all solutions in the original equation is essential.

Domain Restrictions

When dealing with square roots, the expression inside the root must be non-negative to produce real numbers. This restriction limits the possible values of the variable, so determining the domain before solving helps avoid invalid solutions and simplifies the solving process.

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 their validity and discard any extraneous solutions.