Quadratic Equations

A quadratic equation is a second-order polynomial equation in a single variable with a format of ax² + bx + c = 0, where x represents an unknown, and a, b, and c are constants with a ≠ 0.
The quadratic form is named for the term involving the square of the unknown variable, x².
Quadratic equations can be resolved via three methods - factoring, using the quadratic formula, or by completing the square.
Factoring involves rearranging the expression into two brackets and setting each equal to zero (i.e., if (x+a)(x+b)=0, then x=-a or x=-b).
The quadratic formula is x = [-b ± sqrt(b²-4ac)] / 2a. This formula provides the solutions to any quadratic equation.
Completing the square involves rearranging the quadratic equation into the format “(x+a)² = b”. This is most commonly used to determine the vertex of the parabola.
Real solutions to a quadratic equation are represented by the x-intercepts of the graph of that function. If there are no real solutions, the graph does not touch the x-axis.
If the solutions to a quadratic equation are the same value, this is a repeated root and visually, the graph of the function just touches the x-axis. The graph forms a parabola and touches the x-axis.
The sign of the leading coefficient ‘a’ in the equation determines the direction of the parabola (upwards if a>0, downwards if a<0).
The vertex of a parabola, given by the coordinates (h, k), can be calculated using the formula h = -b/2a, and k = c – b²/4a.
In graphs of quadratic functions, the line of symmetry is the vertical line that passes through the vertex of the parabola.
Quadratics underly many aspects of real-world phenomena, including physics motion problems and optimization problems in various fields.