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.