Solve each equation. See Examples 4–6. ∜(x²+2x)= ∜3

Verified step by step guidance

1. Recognize that the equation involves fourth roots (the fourth root symbol ∜). The equation is \( \sqrt[4]{x^2 + 2x} = \sqrt[4]{3} \).
2. Since the fourth roots on both sides are equal, you can set the radicands (the expressions inside the roots) equal to each other: \( x^2 + 2x = 3 \).
3. Rewrite the equation as a quadratic by moving all terms to one side: \( x^2 + 2x - 3 = 0 \).
4. Factor the quadratic equation if possible, or use the quadratic formula to find the values of \( x \). The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -3 \).
5. Solve for \( x \) using the factored form or quadratic formula, then check your solutions in the original equation to ensure they do not produce extraneous results (since we are dealing with even roots).

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 fourth root of an expression, which must be handled carefully.

Properties of Exponents and Roots

The fourth root of a number is equivalent to raising that number to the power of 1/4. To eliminate the root, both sides of the equation can be raised to the fourth power, which simplifies the equation to a polynomial form. This property helps in transforming the radical equation into a solvable algebraic equation.

Solving Quadratic Equations

After removing the fourth root, the resulting equation is quadratic in form (x² + 2x = 3). Solving quadratic equations involves rearranging terms, factoring, or using the quadratic formula to find the values of x. Checking solutions in the original equation is important to avoid extraneous roots.