Page 2: Equations Reducible to Quadratic Form

This page explores equations that can be transformed into quadratic equations, focusing on biquadratic equations.

Definition: A biquadratic equation has the form ax⁴ + bx² + c = 0, where a, b, and c are constants and a ≠ 0.

To solve biquadratic equations:

1. Substitute t = x²
2. Solve the resulting quadratic equation in t
3. Solve x = ±√t for each solution of t

Example: Solve x⁴ - 3x² + 2 = 0

1. Let t = x²
2. Solve t² - 3t + 2 = 0
3. Find t₁ = 2 and t₂ = 1
4. Solve x² = 2 and x² = 1
5. Final solutions: x = ±√2 and x = ±1

This method demonstrates how complex equations can be simplified and solved using quadratic equation techniques.

Page 3: Quadratic Inequalities

This page introduces the concept of quadratic inequalities and outlines a method for solving them.

Definition: A quadratic inequality is an inequality that involves a quadratic expression and one of the symbols <, ≤, >, or ≥.

The solution to a quadratic inequality is typically an interval or union of intervals on the real number line.

Method for solving quadratic inequalities:

1. Move all terms to one side of the inequality sign
2. Treat the left side as a quadratic function
3. Graph the quadratic function
4. Read the solution from the graph, considering the inequality sign

Highlight: The graph of a quadratic function is a parabola, which is crucial for visualizing and solving quadratic inequalities.

Understanding the relationship between the graph and the inequality sign is key to correctly interpreting the solution.

Page 4: Graphing Quadratic Functions

This page focuses on the graphical representation of quadratic functions and their relationship to the x-axis.

Vocabulary: The roots of a quadratic equation are the x-intercepts of its corresponding quadratic function graph.

The page presents several cases for the position of a parabola relative to the x-axis:

• Parabola intersects the x-axis at two points (two real roots)
• Parabola touches the x-axis at one point (one real root, Δ = 0)
• Parabola does not intersect the x-axis (no real roots)

Highlight: The discriminant (Δ) determines the nature of the roots and the position of the parabola.

Vieta's formulas are introduced, relating the sum and product of roots to the coefficients of the quadratic equation:

• Sum of roots: x₁ + x₂ = -b/a
• Product of roots: x₁ · x₂ = c/a

Example: For ax² + bx + c = 0,

• If x₁ + x₂ > 0, the parabola intersects the positive x-axis
• If x₁ · x₂ < 0, the roots have opposite signs

These relationships help in analyzing the behavior of quadratic functions without solving the equation explicitly.

Page 1: Solving Quadratic Equations

This page introduces the basics of quadratic equations and provides step-by-step solutions to example problems.

Definition: A quadratic equation has the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Quadratic equations can have one, two, or no real solutions. To solve them, we calculate the discriminant (Δ) using the formula Δ = b² - 4ac.

Example: Solve 3x² - 7x + 2 = 0

1. Calculate Δ = $-7$² - 4(3)(2) = 49 - 24 = 25
2. Use the quadratic formula: x = $-b ± √Δ$ / (2a)
3. x₁ = $-(-7) + √25$ / (2(3)) = $7 + 5$ / 6 = 2
4. x₂ = $-(-7) - √25$ / (2(3)) = $7 - 5$ / 6 = 1/3

Highlight: Always check your solutions by substituting them back into the original equation.

The page also includes another example of solving a quadratic equation with fractional coefficients, demonstrating the versatility of the quadratic formula.