Solving Equations by Factoring

Simple Factoring

Factoring is a method used to solve polynomial equations by expressing the equation as a product of its factors and setting each factor equal to zero. This technique is especially useful for equations where all terms can be moved to one side, resulting in zero on the other side.

• Key Point 1: Move all terms to one side of the equation so that the equation equals zero.

• Key Point 2: Factor the polynomial completely.

• Key Point 3: Set each factor equal to zero and solve for the variable.

• Example: Solve

  • Move all terms to one side:

  • Factor:

  • Further factor:

  • Set each factor to zero: , ,

  • Solutions:

• Factoring by Grouping: Used when a polynomial has four terms. Group terms to factor common elements.

• Example: Solve

  • Group:

  • Factor:

  • Set each factor to zero and solve.

Additional info: Factoring is foundational for solving higher-degree polynomial equations and is often the first step before applying other algebraic techniques.

Radical Equations

Solving Equations Involving Square Roots

Radical equations are equations in which the variable appears under a radical sign, most commonly a square root. Solving these equations often requires isolating the radical and then raising both sides to a power to eliminate the radical.

• Key Point 1: Isolate the radical expression on one side of the equation.

• Key Point 2: Raise both sides to the appropriate power to eliminate the radical.

• Key Point 3: Solve the resulting equation for the variable.

• Key Point 4: Check for extraneous solutions by substituting answers back into the original equation, since squaring both sides can introduce invalid solutions.

• Example: Solve

  • Isolate the radical:

  • Square both sides:

  • Expand:

  • Rearrange:

  • Factor:

  • Solutions:

  • Check both in the original equation to confirm validity.

Additional info: Always check for extraneous solutions when solving radical equations.

Equations with Rational Exponents

Solving Equations Involving Fractional Powers

Equations with rational exponents involve variables raised to fractional powers. These can be solved by isolating the term with the rational exponent and then raising both sides to the reciprocal of the exponent.

• Key Point 1: Isolate the term with the rational exponent.

• Key Point 2: Raise both sides to the reciprocal of the exponent to eliminate the fractional power.

• Key Point 3: If the denominator of the exponent is even, check for extraneous solutions (since even roots have domain restrictions).

• Example: Solve

  • Isolate:

  • Divide:

  • Raise both sides to the power:

Additional info: Rational exponents can be rewritten as roots and powers, e.g., .

Equations that are Quadratic in Form

Solving Non-Quadratic Equations Using Substitution

Some equations, while not quadratic, can be transformed into quadratic equations by substituting a new variable for a power of the original variable. This method is useful for equations where the exponents are multiples of each other.

• Key Point 1: Identify a substitution that will convert the equation into a quadratic form (e.g., let or ).

• Key Point 2: Solve the resulting quadratic equation for the new variable.

• Key Point 3: Substitute back to solve for the original variable.

• Example: Solve

  • Let , so

  • Factor:

  • So or

  • Back-substitute: ; (no real solution)

Additional info: This technique is called "quadratic in form" substitution and is widely used for higher-degree polynomials.

Absolute Value Equations

Solving Equations Involving Absolute Value

Absolute value equations involve expressions within absolute value bars, which represent the distance from zero and are always non-negative. To solve these equations, isolate the absolute value and then split into two cases.

• Key Point 1: Isolate the absolute value expression on one side of the equation.

• Key Point 2: Set up two equations: one where the expression inside the absolute value equals the positive value, and one where it equals the negative value.

• Key Point 3: Solve both equations for the variable.

• Example: Solve

  • Isolate:

  • Set up two equations: and

  • Solve: and

Additional info: Always check solutions in the original equation, especially if the absolute value is set equal to a negative number (which would have no solution).