Precalculus Test 1 Review: Linear, Quadratic, Polynomial, and Miscellaneous Equations

Test 1 Review: Key Precalculus Topics

I. Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the first power. They are fundamental in algebra and serve as the basis for more advanced topics.

• Definition: An equation of the form $ax + b = 0$, where $a \neq 0$.

• Solution: Isolate the variable to find $x = -\frac{b}{a}$.

• Example: Solve $2x + 3 = 7$. $2x = 4$ $x = 2$

II. Quadratic Equations

Quadratic equations are second-degree equations, typically written as $ax^2 + bx + c = 0$.

• Standard Form: $ax^2 + bx + c = 0$

• Solution Methods:

  • Factoring

  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

  • Completing the Square

• Example: Solve $x^2 - 5x + 6 = 0$ by factoring. $(x-2)(x-3) = 0$ $x = 2$ or $x = 3$

III. Miscellaneous Equations

This section covers equations that do not fit strictly into linear or quadratic categories, including rational, radical, and equations solved by substitution.

A. Rational Equations

• Definition: Equations involving fractions with polynomials in the numerator and denominator.

• Example: Solve $\frac{2}{x} = 4$. $2 = 4x$ $x = \frac{1}{2}$

B. Radical Equations

• Definition: Equations involving roots, such as square roots.

• Example: Solve $\sqrt{x+3} = 5$. $x+3 = 25$ $x = 22$

C. Using Substitution

• Method: Substitute one variable or expression for another to simplify and solve equations.

• Example: If $y = 2x$, substitute into $x + y = 9$ to get $x + 2x = 9 \Rightarrow x = 3$.

IV. Linear Systems

Linear systems involve solving two or more linear equations simultaneously to find the values of variables that satisfy all equations.

• Methods:

  • Substitution

  • Elimination

  • Graphical Solution

• Example: Solve the system: $x + y = 5$ $x - y = 1$ Add: $2x = 6 \Rightarrow x = 3$ Substitute: $3 + y = 5 \Rightarrow y = 2$

V. Polynomial Equations of Higher Degree

These are equations where the highest power of the variable is greater than two. They may require special techniques for solution.

• Listing Rational Solutions: Use the Rational Root Theorem to list all possible rational roots.

• Solving Over Complex Numbers: Some equations have complex (non-real) solutions, found using factoring, synthetic division, or the quadratic formula for reducible cases.

• Example: Solve $x^3 - 1 = 0$. $x^3 = 1$ $x = 1$ (real root), $x = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$ (complex roots)

VI. Applications

Application problems use algebraic equations to model and solve real-world scenarios.

A. Mixture Problems

• Definition: Problems involving combining substances of different concentrations to achieve a desired mixture.

• Example: How much of a 10% solution should be mixed with a 30% solution to get 20 liters of a 20% solution?

B. Motion Problems

• Definition: Problems involving distance, rate, and time, typically using the formula $d = rt$.

• Example: If a car travels at 60 mph for 2 hours, how far does it go? $d = 60 \times 2 = 120$ miles