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Algebraic Equations Overview

Sep 3, 2025

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Overview

This lecture covers solving various types of algebraic equations, including linear, rational, and quadratic equations, with emphasis on techniques like distribution, factoring, and the quadratic formula.

Understanding Equations

  • An equation is a statement that two algebraic expressions are equal.
  • Solving means isolating the variable (e.g. x) on one side and numbers on the other.
  • An equation with one solution is called a conditional equation.
  • An identity is an equation that is true for all values (e.g., 1 = 1).

Solving Linear Equations Example

  • Distribute coefficients to eliminate parentheses on both sides.
  • Combine like terms, isolate variable terms on one side and constants on the other.
  • Divide by the coefficient of the variable to solve for the variable.
  • Example solution: For -3(4t-5) = 5(6-2t), the solution is t = -15/2.

Solving Equations with Fractions

  • Multiply both sides by the least common denominator (LCD) to clear fractions.
  • Simplify and solve the resulting linear equation.
  • Always check if the solution causes division by zero in denominators.
  • Example: For (2x+2)/(x-2) = 1, solution is x = -4.

Solving Quadratic Equations by Factoring

  • Factor out the greatest common factor (GCF) if possible.
  • Use the difference of squares: a²-b² = (a-b)(a+b).
  • Set each factor equal to zero and solve for the variable.
  • Example: For x³-4x=0, solutions are x = 0, 2, -2.

Solving Equations with Rational Expressions

  • Multiply through by LCD to eliminate denominators.
  • Reduce to a quadratic equation and solve by factoring.
  • Example: For 1-9/x²=0, solutions are x = 3, -3.

Solving Quadratic Equations with the Quadratic Formula

  • Multiply both sides by LCD if necessary.
  • Expand and rearrange into standard quadratic form: ax²+bx+c=0.
  • If not factorable, use the quadratic formula: x = [-b ± √(b²-4ac)]/(2a).
  • Example: For x+8 = 1/(x-4), solutions are x = -2 ± √37.

Key Terms & Definitions

  • Equation — A mathematical statement showing that two expressions are equal.
  • Identity — An equation true for all values of the variable.
  • Conditional Equation — An equation true for specific values of the variable.
  • Least Common Denominator (LCD) — The smallest common multiple of denominators in a rational equation.
  • Greatest Common Factor (GCF) — The largest expression that can be factored from terms.
  • Difference of Squares — A factoring form: a²-b² = (a-b)(a+b).
  • Quadratic Formula — Formula to solve ax²+bx+c=0: x = [-b ± √(b²-4ac)]/(2a).

Action Items / Next Steps

  • Practice solving equations using distribution, factoring, and the quadratic formula.
  • Always check solutions for extraneous answers, especially in rational equations.
  • Review definitions and examples before attempting homework problems.

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