Quadratic Equations

A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a is not zero. The highest power of x is 2, which is why the graph bends into a parabola instead of a straight line. A solution of the equation is called a root; it is an x-value that makes the equation true.

The roots of ax² + bx + c = 0 are exactly the points where the parabola y = ax² + bx + c crosses the x-axis, because there y = 0. A parabola can cross the axis at two points, just touch it at one point, or miss it entirely. That picture is the whole topic in one image.

Factorisation splits the quadratic into two brackets and uses the zero-product rule. Completing the square reshapes it into (x + p)² = q. The quadratic formula works every time and is just completing the square done once for the general case. Pick factorisation when roots are simple; reach for the formula when they are not.

The part under the square root, D = b² − 4ac, settles everything before you finish. If D > 0 there are two different real roots; if D = 0 there is one repeated root (the curve just touches the axis); if D < 0 there are no real roots (the curve misses the axis). You can classify a quadratic with a single subtraction.