Radical Equations and Equations Quadratic in Form: Methods and Examples

Radical Equations and Equations Quadratic in Form

Introduction to Radical Equations

Radical equations are equations in which the variable appears under a radical sign, such as a square root, cube root, or higher-order root. Solving these equations often requires isolating the radical and then raising both sides of the equation to a power that eliminates the radical.

• Radical Equation: An equation in which the variable is inside a radical, such as $\sqrt{x}$ or $\sqrt[3]{x}$.

• Key Steps: Isolate the radical, raise both sides to the appropriate power, solve, and check for extraneous solutions.

Solving Radical Equations

To solve a radical equation, follow these systematic steps:

1. Isolate the radical expression on one side of the equation.

2. Raise both sides of the equation to the power that matches the index of the radical (e.g., square both sides for a square root).

3. Solve the resulting equation for the variable.

4. Check all potential solutions in the original equation, as extraneous solutions may arise from the process of raising both sides to a power.

Example 1: Solve $\sqrt{x - 3} = 5$

• Step 1: Isolate the radical: $\sqrt{x - 3} = 5$

• Step 2: Square both sides: $(\sqrt{x - 3})^2 = 5^2$

• Step 3: $x - 3 = 25 \implies x = 28$

• Step 4: Check: $\sqrt{28 - 3} = \sqrt{25} = 5$ (valid solution)

Why Check Solutions? Raising both sides to a power can introduce extraneous solutions that do not satisfy the original equation. Always substitute solutions back into the original equation to verify their validity.

Equations Quadratic in Form

Some equations, though not written as standard quadratics, can be transformed into quadratic equations by substitution. These are called equations quadratic in form.

• Quadratic in Form: An equation that can be rewritten as $au^2 + bu + c = 0$ by substituting a variable or expression for $u$.

• Method: Substitute $u$ for the repeated expression, solve the quadratic equation, then substitute back and solve for the original variable.

Example 2: Solve $(x^2 - 3x)^2 + 5(x^2 - 3x) - 6 = 0$

• Let $u = x^2 - 3x$

• Equation becomes $u^2 + 5u - 6 = 0$

• Factor: $(u + 6)(u - 1) = 0 \implies u = -6$ or $u = 1$

• Substitute back: $x^2 - 3x = -6$ and $x^2 - 3x = 1$

• Solve each resulting quadratic equation for $x$

Solving Equations by Factoring

Factoring is a fundamental method for solving polynomial equations, including those that are not strictly quadratic. The process involves rewriting the equation in a form where one side is zero and the other is a product of factors.

• Factoring: Expressing a polynomial as a product of its factors.

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

Example 3: Solve $x^4 + 4x^2 - 5x = 36$

• Rewrite: $x^4 + 4x^2 - 5x - 36 = 0$

• Factor the polynomial (if possible) and set each factor equal to zero.

• Solve for $x$.

Summary Table: Methods for Solving Equations

Equation Type	Method	Key Steps
Radical Equation	Isolate radical, raise to power, solve, check	Isolate, exponentiate, solve, verify
Quadratic in Form	Substitution, solve quadratic, back-substitute	Let $u =$ expression, solve for $u$, solve for variable
Polynomial (Factorable)	Factoring	Rewrite as zero, factor, apply zero product property

Additional info: Checking solutions is especially important for radical equations due to the possibility of extraneous solutions introduced by raising both sides to a power.