College Algebra

College Algebra covers fundamental algebraic concepts, including functions, equations, and inequalities, essential for higher-level mathematics.

Algebraic Expressions and OperationsSolving Equations and InequalitiesFunctions and Their Properties
Quadratic Equations and ApplicationsExponential and Logarithmic FunctionsSystems of Equations
Budgeting and Personal FinanceScience and Engineering ModelsBusiness and Economics
Practice and Review RegularlyBreak Down Complex Problems
Algebraic Expression...Solving Equations an...Functions and Their ...Quadratic Equations ...Exponential and Loga...Systems of EquationsBudgeting and Person...Science and Engineer...Business and Economi...Practice and Review ...Break Down Complex P...
Advanced Topics

Quadratic Equations and Applications

Exploring Quadratic Equations

Quadratic equations have the form \(ax^2 + bx + c = 0\). They appear everywhere—from physics to finance.

Solving Methods

  • Factoring: Find two numbers that multiply to \(ac\) and add to \(b\).
  • Quadratic Formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
  • Completing the Square: Rearrange and solve by taking square roots.

Real-World Applications

  • Projectile motion (sports, physics)
  • Optimizing area or profit

Visualizing Quadratics

The graph is a parabola—its vertex shows the maximum or minimum value.

Key Formula

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Examples

  • Solve \(x^2 - 5x + 6 = 0\) by factoring: \((x-2)(x-3) = 0\), so \(x=2\) or \(x=3\).

  • A ball thrown upward follows \(h(t) = -16t^2 + 32t + 5\); its height at any time \(t\) is found by solving this quadratic.

In a Nutshell

Quadratic equations solve problems with squared variables and appear in many real-life contexts.

Key Terms

Discriminant
The value \(b^2 - 4ac\) in the quadratic formula, which determines the number and type of solutions.
Vertex
The highest or lowest point of a parabola.
Next
Exponential and Logarithmic Functions