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6: Solving Equations

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    173414

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    • 6.1: Solving Literal Equations
      This page covers fundamental concepts of equations, including definitions and types such as conditional equations and identities. It explains solving techniques for isolating variables using properties of equality, while emphasizing the treatment of other variables as constants. The page provides guidelines for reversing operations and includes examples that illustrate solving equations in different contexts, while also warning against common errors like division by zero and factor handling.
    • 6.2: Solving Linear Equations - One or Two-Steps
      This page offers a comprehensive overview of solving linear equations in one variable. It emphasizes isolating the variable using properties of equality and defines key concepts such as linear equations, solutions, and equivalent equations. The document explains how to manipulate equations through addition, subtraction, multiplication, and division to achieve this isolation.
    • 6.3: Solving Linear Equations - Multistep with Variables on One Side
      This page introduces the basics of solving linear equations in one variable, defining essential concepts and the form \(ax + b = 0\). It emphasizes the importance of like terms and the distributive property for simplification. Key properties of equality are discussed for effective manipulation. The page provides a step-by-step strategy for isolating variables, supported by examples that demonstrate distribution, combining like terms, and managing negatives and fractions.
    • 6.4: Solving Linear Equations - Multistep with Variables on Both Sides
      This page explains solving linear equations in one variable by gathering terms on one side before isolating the variable. It defines linear equations and equivalent equations, highlighting operations that maintain equivalence and potential outcomes: one solution, infinitely many, or none. Tips and examples are provided, along with common error warnings. Additionally, it discusses a case where terms cancel, indicating all real numbers are solutions.
    • 6.5: Solving Absolute Value Equations
      This page introduces absolute value equations, defining absolute value as a distance from zero and detailing solving techniques through case analysis. It emphasizes key theorems regarding equations like |x|=c, highlighting the importance of checking for extraneous solutions. A specific example, \(2x-1=-(x+4)\), is solved step-by-step, yielding \(x=-1\) and \(x=5\), which are verified by substitution.
    • 6.6: Solving Quadratic Equations by Factoring
      This page covers quadratic equations, defining their standard form and introducing the Zero Product Property for solving them via factoring. It outlines a three-step strategy: rewriting in standard form, factoring fully, and using the Zero Product Property. The text includes examples addressing different scenarios like varying leading coefficients, perfect squares, and common factors, while advising against dividing by variables to avoid losing solutions.
    • 6.7: Solving Quadratic Equations Using Extraction of Roots
      This page explains the extraction of roots as a method for solving quadratic equations without linear terms. It details how this technique yields two solutions for positive constants, one for zero, and none for negatives. Emphasis is placed on isolating the squared term before applying the square root property, supported by examples that demonstrate solving perfect squares and binomials.
    • 6.8: Solving Quadratic Equations by Completing the Square
      This page covers solving quadratic equations by completing the square, detailing the transformation into a perfect square trinomial for root extraction. It defines key terms, presents a step-by-step method, and introduces important theorems, such as the Perfect Square Construction and Square Root Property. Common pitfalls are highlighted, and multiple examples demonstrate the technique across different types of quadratics, including those with irrational roots and no real solutions.
    • 6.9: Solving Quadratic Equations Using the Quadratic Formula
      This page covers quadratic equations in standard form \(ax^2 + bx + c = 0\) and explains the discriminant \(\Delta = b^2 - 4ac\), which indicates the nature of the roots. The quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\) is introduced for finding roots. Examples illustrate different solutions, including real and complex scenarios, through specific equations, emphasizing discriminant analysis. Sources cited support the content.
    • 6.10: Applications Involving Quadratic Equations
      This page covers the use of quadratic equations in real-world scenarios like projectile motion and geometry, focusing on modeling and solving related problems while discarding invalid solutions. It defines key concepts and illustrates applications through examples, including a model rocket launched from a height, where the quadratic formula helps determine its impact time, found to be approximately 5.07 seconds after launch.
    • 6.11: Solving Polynomial Equations
      This page explains how to solve polynomial equations of degree three or higher, introducing key concepts like polynomial definitions, roots, and essential theorems (Zero-Product, Factor, Rational Root). It emphasizes finding rational roots through the Rational Root Theorem and properly applying the Factor Theorem while avoiding division by variables.
    • 6.12: Solving Rational Equations
      This page explains rational equations, emphasizing the importance of using the least common denominator (LCD) for solving them. It defines key concepts like extraneous solutions and the necessity to check for values that make denominators zero. The solving process includes factoring, multiplying by the LCD, simplifying to a polynomial equation, and verifying solutions against restrictions. Multiple examples are provided to demonstrate the method.
    • 6.13: Solving Radical Equations
      This page covers radical equations, focusing on definitions, principal roots, and the critical need to check for extraneous solutions after squaring. It discusses the limitations of the Power Property of Equality when dealing with even powers and provides detailed steps for solving radical equations, supplemented with examples.
    • 6.14: Solving Exponential Equations
      This page discusses exponential equations and logarithms, detailing their definitions and inverse relationship. It emphasizes the One-to-One Property and the Power Rule for Logarithms for solving equations. Various examples demonstrate methods to solve exponential equations using both common bases and logarithmic techniques.
    • 6.15: Solving Logarithmic Equations
      This page covers logarithmic equations, explaining their solutions through logarithmic and exponential forms, and defining key laws such as the product, quotient, and power rules. It highlights the necessity of isolating logarithms prior to conversion and the importance of validating solutions to avoid negative arguments.
    • 6.16: Solving Equations Quadratic-in-Form
      This page covers quadratic-in-form equations represented as \(au^2 + bu + c = 0\), where \(u\) is an expression in \(x\). It highlights the necessity of substituting back to determine \(x\) and warns against halting at the \(u\) solutions. Through various examples, including biquadratic equations and those involving radical and rational substitutions, it illustrates the solving process and concludes with a method for finding real and complex solutions.

    This page titled 6: Solving Equations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Roy Simpson, Cosumnes River College.