Algebra Class 10: Quadratic Equations

Definition and Basic Form

• A quadratic equation is a polynomial equation of the second degree.
• It's typically written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not zero.
• The variable 'x' represents an unknown value.

Standard Form

• The standard form of a quadratic equation is ax² + bx + c = 0.
• 'a', 'b', and 'c' are coefficients, and they can be any real numbers.
• 'x' is the variable.

Solving Quadratic Equations

• Several methods exist to solve quadratic equations.
• Factoring: If factorable, rewrite as (px + q)(rx + s) = 0. Solve for x.
• Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a. Solves any quadratic equation.
• Completing the square: Rewrites the equation into a perfect square trinomial.
• Graphing: Plot the parabola y = ax² + bx + c and find the x-intercepts.

Discriminant

• The discriminant is the part of the quadratic formula under the square root: b² - 4ac.
• It indicates the nature of the quadratic equation's roots.
  • Positive discriminant: Two distinct real roots.
  • Zero discriminant: One real root (or two identical roots).
  • Negative discriminant: No real roots; two complex roots.

Roots (Solutions)

• Roots (solutions) are the values of 'x' satisfying the equation.
• They represent the x-intercepts of the quadratic function's graph.

Properties of Parabolas

• The graph of a quadratic equation is a parabola.
• Parabola opens upwards if 'a' is positive, downwards if 'a' is negative.
• The vertex is the maximum or minimum point.
• The axis of symmetry is a vertical line through the vertex.

Vertex of a Parabola

• The vertex is where the parabola changes from increasing to decreasing (or vice versa).
• The x-coordinate of the vertex is -b / 2a.

Applications of Quadratic Equations

• Quadratic equations model various scenarios, including:
  • Projectile motion calculations.
  • Finding maximum or minimum areas of shapes.
  • Modeling growth or decay processes.

Example Problems

• Examples using factoring, the quadratic formula, or graphing will demonstrate these concepts.

Types of Roots

• The discriminant (b² - 4ac) determines the nature of the roots.
  • Real and distinct: Two different real number roots.
  • Real and equal: One real number root (repeated root).
  • Imaginary (complex): Two complex (imaginary) roots.

This quiz covers the definition and basic forms of quadratic equations, including methods for solving them such as factoring, using the quadratic formula, and completing the square. Test your understanding of the discriminant and graphing techniques for quadratic equations.