Solve each equation. ∜(x²+2x)= ∜3

Verified step by step guidance

1. Recognize that the equation involves fourth roots (the fourth root symbol ∜ means raising to the power of 1/4). The equation is given as \(\sqrt[4]{x^{2} + 2x} = \sqrt[4]{3}\).
2. Since both sides are fourth roots, you can eliminate the roots by raising both sides of the equation to the 4th power. This gives: \((\sqrt[4]{x^{2} + 2x})^{4} = (\sqrt[4]{3})^{4}\).
3. Simplify both sides after raising to the 4th power: \(x^{2} + 2x = 3\).
4. Rewrite the equation in standard quadratic form by subtracting 3 from both sides: \(x^{2} + 2x - 3 = 0\).
5. Solve the quadratic equation \(x^{2} + 2x - 3 = 0\) using factoring, completing the square, or the quadratic formula to find the values of \(x\).

Key Concepts

Radical Equations and Roots

Radical equations involve variables inside root expressions, such as square roots or fourth roots. Understanding how to isolate and manipulate these roots is essential for solving the equation. In this problem, the fourth root (∜) indicates the variable expression is raised to the 1/4 power.

Eliminating Radicals by Raising to a Power

To solve equations with radicals, both sides can be raised to the power that corresponds to the root to eliminate the radical. For a fourth root, raising both sides to the 4th power removes the root, simplifying the equation to a polynomial form that is easier to solve.

Solving Quadratic Equations

After removing the radical, the resulting equation is often quadratic, involving terms like x² and x. Solving quadratic equations requires methods such as factoring, completing the square, or using the quadratic formula to find the values of x that satisfy the equation.