Quadratics
Quadratic Equations and Functions
- A quadratic equation takes the form ax^2 + bx + c = 0, where a ≠ 0.
- The solutions to a quadratic equation are given by the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a).
- The "discriminant" is the term b^2 - 4ac. Its value determines the number of roots the equation has.
- The graph of a quadratic function is called a parabola. It is symmetric about the line x = -b/2a, known as the axis of symmetry.
- The lowest or highest point on the graph of a quadratic function is called the vertex. For a function given in the form y = ax^2 + bx + c, the vertex occurs at x = -b/2a.
- The y-intercept of a quadratic function is the value of c (since y = c when x = 0).
- The x-intercepts or roots of a quadratic function are the solutions to the equation ax^2 + bx + c = 0.
Factoring Quadratics
- Factoring is a method used to solve a quadratic equation. An equation can be factored if it can be rewritten in the form (dx + e)(fx + g) = 0, where d and f are the factors of a, and e and g are the factors of c that add up to b.
- The zero-product property states that if a product of factors equals zero, then at least one of the factors must be zero. This property allows us to set each factor equal to zero and solve for x.
- Factoring is not always possible, particularly in cases where the discriminant is negative, expressing that the equation has no real roots.
Quadratic Inequalities
- Quadratic inequalities take the form ax^2 + bx + c < 0, ax^2 + bx + c ≤ 0, ax^2 + bx + c > 0, or ax^2 + bx + c ≥ 0.
- Solve a quadratic inequality by first factoring the quadratic function and then determining the intervals where the function is positive or negative.
- Plot the graph; where the graph lies above the x-axis, y > 0, and where the graph lies below the x-axis, y < 0.
Completing the Square
- Completing the square is another method used to solve a quadratic equation, especially useful when the equation does not factor easily.
- This process involves transforming the given equation into the form (x - h)^2 = k, which allows us to identify the vertex 'h' and y-intercept 'k'.
- After completing the square and simplifying the equation, solutions can be found by solving x - h = sqrt(k) and x - h = -sqrt(k).
Quadratic Formula Derivation
- The quadratic formula can be derived from the process of completing the square
- Start with the standard form of a quadratic equation (ax^2 + bx + c = 0), and manipulate it to derive the formula: *x = [-b ± sqrt(b^2 - 4ac)] / 2a.