Solve each equation. √(2√(7x+2)) = √(3x+2)

Verified step by step guidance

1. Start with the given equation: \(\sqrt{2\sqrt{7x+2}} = \sqrt{3x+2}\).
2. Square both sides of the equation to eliminate the outer square roots: \(\left(\sqrt{2\sqrt{7x+2}}\right)^2 = \left(\sqrt{3x+2}\right)^2\) which simplifies to \(2\sqrt{7x+2} = 3x + 2\).
3. Isolate the remaining square root term: \(2\sqrt{7x+2} = 3x + 2\) implies \(\sqrt{7x+2} = \frac{3x + 2}{2}\).
4. Square both sides again to remove the square root: \(\left(\sqrt{7x+2}\right)^2 = \left(\frac{3x + 2}{2}\right)^2\) which simplifies to \(7x + 2 = \frac{(3x + 2)^2}{4}\).
5. Multiply both sides by 4 to clear the denominator and then expand and simplify the resulting quadratic equation. After that, solve for \(x\) and check for extraneous solutions by substituting back into the original equation.

Key Concepts

Properties of Square Roots

Square roots represent the principal non-negative root of a number. Understanding how to manipulate and simplify nested square roots, such as √(2√(7x+2)), is essential. This includes knowing that √a = √b implies a = b when both sides are non-negative.

Solving Radical Equations

Solving equations involving radicals often requires isolating the radical expression and then squaring both sides to eliminate the square root. Care must be taken to check for extraneous solutions introduced by squaring.

Domain Restrictions

Since square roots are only defined for non-negative radicands, it is important to determine the domain of the variable by setting the expressions inside the radicals greater than or equal to zero. This ensures solutions are valid within the problem's context.