The Geometry of the Relationships Between the Coefficients and the Solutions of a Quadratic Equation

Autor:Simona Riva

Thema:Algebra, Gleichungen, Hyberbel, Geraden, Quadratische Gleichungen, Sekante, Tangente

Quadratic Equations, Coefficients and Solutions

Given a quadratic equation in standard form

, with

, we can relate the coefficients

to the solutions

and

of the equation:

the sum of the solutions is
the product of the solutions is

Let Us Visualize First...

Interact with the app below. In the first part, you will discover how the dynamic visualization in the following part is obtained. Use the sliders to explore dynamically the relationship between coefficients and solutions of a quadratic equation. The values displayed in the equations representing the sum and product of the solutions are expressed in exact rational form, while the solutions of the equation are expressed in decimal form, rounded to the nearest hundredth.

Conclusions

Have you discovered the relationship between the coefficients of the equation and the coordinates of the points and ? Explain your conclusions.

Exploration 1

Solve the equation without using the app above. How do you expect the relative positions of the corresponding hyperbola and line be? Intersecting, tangent or disjoint? Explain the reasoning behind your answer and verify your conjecture using the app above.

Exploration 2

Solve the equation using the app above. Observe the relative positions of the corresponding hyperbola and line. How can you connect these relative positions to the value of the discriminant of the equation? Explain the reasoning behind your answer and verify your conjecture by calculating the of the equation.

Exploration 3

Consider the equation . If you apply the formulas that represent in geometrical form the relationship between the coefficients and the solutions of the equation, you obtain that the product of the solutions is represented by the equation . Does this equation still represent a rectangular hyperbola in the Cartesian plane? How would you apply the zero product property to solve the equation and to understand how to draw the graph of ? Explain the reasoning behind your answers and verify your conjecture using the app above.