College Algebra: Practice Problems and Concepts

Algebraic Equations and Expressions

Solving Radical Equations

Radical equations are equations in which the variable appears under a radical sign. To solve these, isolate the radical and then raise both sides to the appropriate power to eliminate the radical.

• Key Point: Always check for extraneous solutions after solving, as squaring both sides can introduce invalid solutions.

• Example: Solve $\sqrt{2x-1} - \sqrt{x-5} = 3$.

  1. Isolate one radical: $\sqrt{2x-1} = 3 + \sqrt{x-5}$

  2. Square both sides: $2x-1 = (3 + \sqrt{x-5})^2$

  3. Expand and solve for $x$.

Solving Polynomial Equations

Polynomial equations involve variables raised to whole number powers. Solutions can be found by factoring, using the quadratic formula, or other algebraic methods.

• Key Point: For higher-degree polynomials, look for factoring opportunities or use substitution.

• Example: Solve $x^6 - 7x^3 - 8 = 0$.

  • Let $y = x^3$, so the equation becomes $y^2 - 7y - 8 = 0$.

  • Factor: $(y - 8)(y + 1) = 0$

  • So $y = 8$ or $y = -1$, thus $x^3 = 8$ or $x^3 = -1$.

  • Therefore, $x = 2$ or $x = -1$.

Solving Inequalities and Interval Notation

Inequalities are solved similarly to equations, but the solution is a range of values. Interval notation is used to express these solutions compactly.

• Key Point: When multiplying or dividing both sides by a negative number, reverse the inequality sign.

• Example: Solve $2 + |2x - 3| \leq 4$.

  • Isolate the absolute value: $|2x - 3| \leq 2$

  • Write as a compound inequality: $-2 \leq 2x - 3 \leq 2$

  • Solve for $x$: $1/2 \leq x \leq 5/2$

  • Interval notation: $\left[\frac{1}{2}, \frac{5}{2}\right]$

Complex Numbers and Standard Form

Writing Expressions in Standard Form $a + bi$

Complex numbers are written in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$).

• Key Point: Combine like terms and simplify using $i^2 = -1$.

• Examples:

  • a) $(3 - i) + 3i - (5 + 2i)$ Combine real and imaginary parts: $3 - 5 = -2$ (real), $-i + 3i - 2i = 0$ (imaginary) Standard form: $-2$

  • b) $\frac{3}{3 + i}$ Multiply numerator and denominator by the conjugate $(3 - i)$: $\frac{3(3 - i)}{(3 + i)(3 - i)} = \frac{9 - 3i}{9 + 1} = \frac{9 - 3i}{10} = 0.9 - 0.3i$

  • c) $(2 + 5i)^2$ Expand: $4 + 20i + 25i^2 = 4 + 20i - 25 = -21 + 20i$

Quadratic Equations

Solving by Completing the Square

Completing the square is a method to solve quadratic equations by rewriting them in the form $(x + p)^2 = q$.

• Key Point: Move constant to the other side, add $(b/2)^2$ to both sides, then solve for $x$.

• Example: Solve $p^2 + 14p - 38 = 0$ by completing the square.

  1. Move constant: $p^2 + 14p = 38$

  2. Add $(14/2)^2 = 49$ to both sides: $p^2 + 14p + 49 = 87$

  3. Write as square: $(p + 7)^2 = 87$

  4. Solve: $p + 7 = \pm \sqrt{87}$, so $p = -7 \pm \sqrt{87}$

Solving by Quadratic Formula

The quadratic formula solves any quadratic equation $ax^2 + bx + c = 0$:

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

• Example: Solve $4x^2 + 11x - 20 = 0$.

  • $a = 4$, $b = 11$, $c = -20$

  • Discriminant: $11^2 - 4 \cdot 4 \cdot (-20) = 121 + 320 = 441$

  • $x = \frac{-11 \pm \sqrt{441}}{8} = \frac{-11 \pm 21}{8}$

  • So $x = 1.25$ or $x = -4$

Graphing Quadratic Functions

Plotting Points and Analyzing Symmetry

Quadratic functions have the general form $y = ax^2 + bx + c$. Their graphs are parabolas, which may open upward or downward depending on the sign of $a$.

• Key Point: The axis of symmetry is $x = -\frac{b}{2a}$.

• Intercepts:

  • Y-intercept: Set $x = 0$, solve for $y$.

  • X-intercepts: Set $y = 0$, solve for $x$.

• Example: Graph $y = -x^2 + 4$ by plotting points.

  • Vertex: $(0, 4)$

  • Axis of symmetry: $x = 0$

  • Y-intercept: $(0, 4)$

  • X-intercepts: $-x^2 + 4 = 0 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2$

  • Plot points for $x = -2, -1, 0, 1, 2$ and connect to form a downward-opening parabola.

Summary Table: Quadratic Equation Methods

Method	When to Use	Steps
Factoring	When equation can be factored easily	Set equation to zero, factor, set each factor to zero
Completing the Square	When leading coefficient is 1 or easily made 1	Move constant, add $(b/2)^2$, write as square, solve
Quadratic Formula	Any quadratic equation	Identify $a$, $b$, $c$, substitute into formula