Quadratic Equations and Their Solutions

Solving Quadratic Equations

Quadratic equations are equations of the form ax2 + bx + c = 0. There are several methods to solve them:

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Completing the Square: Rewrite the equation in the form (x + p)2 = q and solve for x.

• Quadratic Formula: Use the formula below to find the solutions:

• Discriminant Analysis: The discriminant D = b^2 - 4ac determines the nature of the solutions:

Example: Solve by factoring: .

Quadratic Functions: Graphs and Properties

Vertex, Axis of Symmetry, and Maximum/Minimum

A quadratic function is written as f(x) = ax^2 + bx + c. Its graph is a parabola.

• Vertex: The vertex is the maximum or minimum point. Its x-coordinate is .

• Axis of Symmetry: The vertical line divides the parabola into two symmetric parts.

• Maximum/Minimum Value: If a > 0, the parabola opens upward (minimum); if a < 0, it opens downward (maximum).

Example: For , vertex at , (minimum).

Polynomial Functions

Zeros, Multiplicity, and Graphing

• Zeros: The values of x for which .

• Multiplicity: The number of times a zero is repeated. If the multiplicity is even, the graph touches the x-axis; if odd, it crosses.

Example: has zeros at x = 2 (multiplicity 2), x = -1 (multiplicity 1).

End Behavior and Turning Points

• End Behavior: Determined by the leading term .

• Turning Points: A polynomial of degree n has at most n-1 turning points.

Example: has degree 3, so at most 2 turning points.

Rational Functions

Domain, Asymptotes, and Graphing

• Domain: All real numbers except where the denominator is zero.

• Vertical Asymptotes: Occur at zeros of the denominator (after simplification).

• Horizontal Asymptotes: Determined by the degrees of numerator and denominator:

• Slant (Oblique) Asymptotes: Occur if degree of numerator is exactly one more than denominator.

Example: has vertical asymptotes at and horizontal asymptote at .

Inequalities and Their Graphs

Solving Polynomial and Rational Inequalities

• Polynomial Inequalities: Set the expression to zero, find critical points, and test intervals.

• Rational Inequalities: Set numerator and denominator to zero, find critical points, and test intervals.

Example: Solve . Factor: . Critical points: x = 3, x = -1. Test intervals to find solution set.

Applications and Word Problems

Modeling with Quadratic and Polynomial Functions

• Maximum/Minimum Problems: Use the vertex to find optimal values in real-world contexts (e.g., maximizing area, profit).

• Interpreting Graphs: Use the graph to answer questions about increasing/decreasing intervals, intercepts, and real-world meaning.

Example: A bridge-building problem modeled by a quadratic function; find the number of bridges for maximum profit.

Division of Polynomials

Synthetic and Long Division

• Long Division: Divide polynomials as you would numbers.

• Synthetic Division: A shortcut for dividing by linear factors of the form x - c.

• Remainder Theorem: The remainder of divided by is .

Example: Divide by using synthetic division.

Summary Table: Key Concepts

Additional info: These notes cover topics from Ch. 7 (Quadratic Functions and Equations), Ch. 8 (Polynomial Functions and Rational Functions), and related sections on inequalities and applications, as outlined in the College Algebra curriculum.