Solve radical equations

Learning Objectives Define a radical equation and identify its components. Isolate a radical expression as the first step in solving an equation. Solve radical equations containing one or two radical expressions. Solve radical equations that transform into linear or quadratic equations. Identify and discard extraneous solutions by checking all potential solutions. Apply the process of solving radical equations to contextual problems. Ever wondered how accident investigators calculate a car's speed just from its skid marks? 🚗 They use a formula with a square root, which means they have to solve a radical equation! This tutorial will guide you through the process of solving equations where the variable is trapped inside a radical. You will learn the critical steps of is...

TermDefinitionExample Radical EquationAn equation in which at least one variable expression is located inside a radical (the radicand).√(x + 5) = 3 is a radical equation because the variable 'x' is in the radicand. RadicandThe number or expression inside the radical symbol.In the expression ∛(2x - 7), the radicand is '2x - 7'. Isolating the RadicalThe process of using inverse operations to get the radical term by itself on one side of the equation before eliminating it.To isolate the radical in 2√(x) - 3 = 7, you would add 3 to both sides, then divide by 2 to get √(x) = 5. Principle of PowersA rule stating that if two expressions are equal (a = b), then they will remain equal if raised to the same power (aⁿ = bⁿ). This is the key to eliminating radicals.If √(x) = 4, we...

The Power Rule for Solving Radical Equations If \( \sqrt[n]{A} = B \), then \( (\sqrt[n]{A})^n = B^n \), which simplifies to \( A = B^n \). To eliminate a radical, raise both entire sides of the equation to a power equal to the index of the radical. For a square root (index 2), you square both sides. For a cube root (index 3), you cube both sides. The Mandatory Check Always substitute potential solutions back into the original equation. Because raising both sides of an equation to an even power can introduce extraneous solutions, this step is not optional. You must verify that your final answers make the original equation true.