Friendly Cubics: Visual Proof that x³=bx+c Always Has a Real Root

This GeoGebra example provides an interactive visual proof that every cubic equation of the form x³ = bx + c has at least one real solution. By graphing the cubic y=x³ and the line y=bx+c, the sketch shows that as x approaches positive or negative infinity, the cubic grows faster than any line, forcing the two graphs to intersect at least once. The construction uses sliders for parameters b and c, the Intersect command to compute intersection points, and dynamic text labels to display coordinates. It highlights key concepts such as continuity, the intermediate value theorem, and the geometric interpretation of polynomial equations. Ideal for high school or introductory college mathematics, this sketch helps students connect algebraic root existence with graphical behavior and parameter variation.