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

Test 1 Review: Linear, Quadratic, and Miscellaneous Equations

I. Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the first power. They are foundational in algebra and precalculus, forming the basis for more complex equations and systems.

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

• Solution: $x = -\frac{b}{a}$

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

II. Quadratic Equations

Quadratic equations are second-degree equations, typically written as $ax^2 + bx + c = 0$. They are solved using factoring, completing the square, or the quadratic formula.

• Standard Form: $ax^2 + bx + c = 0$

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

• Example: Solve $x^2 - 5x + 6 = 0$. $(x-2)(x-3) = 0 \implies x = 2, 3$

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: $\frac{1}{x} + \frac{2}{x+1} = 3$

B. Radical Equations

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

• Example: $\sqrt{x+3} = x - 1$

C. Using Substitution

• Method: Substitute one variable or expression to simplify and solve the equation.

• Example: Let $u = x^2$, then solve $u + 2u = 3$.

IV. Linear Systems

Linear systems consist of two or more linear equations solved simultaneously. Solutions can be found using substitution, elimination, or matrix methods.

• Example: Solve the system: $\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$ Solution: $x = 2$, $y = 3$

V. Polynomial Equations of Higher Degree

These are equations of degree three or higher. Solutions may involve factoring, rational root theorem, or synthetic division.

• Example: Solve $x^3 - 6x^2 + 11x - 6 = 0$ Factor: $(x-1)(x-2)(x-3) = 0 \implies x = 1, 2, 3$

• Rational Root Theorem: Used to list all possible rational roots for a polynomial equation.

VI. Applications

Application problems involve translating real-world scenarios into mathematical equations and solving them. Common types include mixture, motion, and simple interest problems.

A. Mixture Problems

• Example: Mixing solutions of different concentrations to achieve a desired concentration.

B. Motion Problems

• Formula: $\text{Distance} = \text{Rate} \times \text{Time}$

C. Simple Interest

• Formula: $I = Prt$ Where $I$ is interest, $P$ is principal, $r$ is rate, and $t$ is time.

Additional info: The assignments referenced correspond to specific practice problems for each topic. Students should review these assignments for targeted practice.