Key Concepts of Quadratic Equations

Quadratic equations are a key part of Algebra and Trigonometry, showing up in various forms like standard and vertex. Understanding their properties, roots, and how to graph them helps solve real-world problems and deepens your math skills.

1. Standard form of a quadratic equation: ax² + bx + c = 0

   • Represents a quadratic equation where 'a', 'b', and 'c' are constants.
   • 'a' cannot be zero; otherwise, it is not a quadratic equation.
   • The graph of this equation is a parabola.

2. Quadratic formula: x = [-b ± √(b² - 4ac)] / 2a

   • Provides a method to find the roots of any quadratic equation.
   • The ± symbol indicates there can be two solutions (roots).
   • Useful when factoring is difficult or impossible.

3. Discriminant: b² - 4ac

   • Determines the nature of the roots of the quadratic equation.
   • If positive, there are two distinct real roots; if zero, one real root; if negative, no real roots.
   • Helps in predicting the number of x-intercepts on the graph.

4. Factoring quadratic equations

   • Involves rewriting the quadratic in the form (px + q)(rx + s) = 0.
   • Useful for quickly finding roots when the equation is factorable.
   • Requires identifying two numbers that multiply to 'ac' and add to 'b'.

5. Completing the square

   • A method to transform a quadratic equation into vertex form.
   • Involves adding and subtracting the same value to create a perfect square trinomial.
   • Useful for deriving the vertex and graphing the parabola.

6. Vertex form of a quadratic equation: y = a(x - h)² + k

   • Highlights the vertex of the parabola at the point (h, k).
   • 'a' determines the direction and width of the parabola.
   • Easier to graph and analyze the properties of the quadratic function.

7. Graphing quadratic functions (parabolas)

   • The graph is a U-shaped curve called a parabola.
   • Opens upwards if 'a' is positive and downwards if 'a' is negative.
   • Key points include the vertex, axis of symmetry, and intercepts.

8. Finding roots/zeros of quadratic equations

   • Roots are the x-values where the graph intersects the x-axis.
   • Can be found using the quadratic formula, factoring, or completing the square.
   • Important for solving real-world problems modeled by quadratic equations.

9. Axis of symmetry: x = -b / (2a)

   • A vertical line that divides the parabola into two mirror-image halves.
   • The x-coordinate of the vertex is located on this line.
   • Helps in graphing the parabola and finding the vertex.

10. Properties of parabolas (concavity, y-intercept, x-intercepts)

    • Concavity is determined by the sign of 'a'; positive means opens up, negative means opens down.
    • The y-intercept is found by evaluating the equation at x = 0 (c).
    • X-intercepts can be found using the quadratic formula or factoring, representing the roots of the equation.