Review of Equations: Linear, Literal, Absolute Value, and Quadratic

Solving Linear Equations

• Steps for solving linear equations:

  • Distribute where necessary.

  • Combine like terms where necessary.

  • Isolate the variable using inverse operations.

• These steps should allow for quick resolution of linear equations.

Literal Equations

• Definition: These involve manipulating a formula to solve for a specific variable. It's essentially using algebraic properties to rearrange an equation.

• Method: Use inverse operations to isolate the desired variable.

• Important Note: Some literal equation problems may require factoring. For example, if the variable you're solving for appears multiple times in the equation, you might need to factor it out before isolating it.

Solving Absolute Value Equations

• Key Steps:

  1. Isolate the absolute value expression first. This means getting the absolute value term by itself on one side of the equation.

  2. Once the absolute value is isolated, create two separate linear equations:

     • One equation sets the expression inside the absolute value equal to the positive value of the number on the other side.

     • The second equation sets the expression inside the absolute value equal to the negative value of the number on the other side.

     • Example: If $|x| = 98$, you would create $x = 98$ and $x = -98$.

  3. Solve both linear equations independently.

• Checking for Extraneous Solutions:

  • It is crucial to check all solutions obtained by plugging them back into the original absolute value equation.

  • An extraneous solution is an answer obtained through correct algebraic steps that, when substituted back into the original equation, does not satisfy the equation. If you get an answer but it doesn't work when plugged in, it's extraneous and should be discarded.

Solving Quadratic Equations

• There are four primary methods for solving quadratic equations:

  1. Factoring: This involves rewriting the quadratic expression as a product of two linear factors and then setting each factor to zero.

  2. Extracting Square Roots: This method is applicable when the quadratic equation can be manipulated into the form $(x-h)^2 = k$. You then take the square root of both sides, remembering both the positive and negative roots.

  3. Completing the Square: This is a technique used to convert a quadratic expression of the form $ax^2 + bx + c$ into an equivalent expression of a perfect square trinomial plus a constant, i.e., $(x-h)^2 + k$. This allows for solving by extracting square roots.

  4. Quadratic Formula: A universal formula that provides the solutions for any quadratic equation in the standard form $ax^2 + bx + c = 0$. The formula is: x = rac{-b ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } rac{ ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ extpm } ext{ } b^2 - 4ac}{2a$$.

• Practice for Today's Lesson: For the current lesson, it is crucial to practice each of these methods as specified in the exercises. While on an exam you might choose your preferred method, today's practice is to ensure familiarity and proficiency with all of them.

• Specific Emphasis on Completing the Square: Most students tend to avoid completing the square. However, practicing it today is important for developing a complete understanding of quadratic equations, even if it won't be explicitly required on a test.

• Quadratic Formula: This is a fundamental method that everyone should know and be proficient with.

General Review Notes

• All review materials, including these topics and supporting notes, are available on the classroom platform.