Determinant questions

Question

What three results does a determinant give us regarding the roots of an equation?

In this case, what would the determinant have to be in order to get each result?

Where does the determinant come from?

How can you prove that every quadratic polynomial has two roots?

Answer 1

You should probably separate your questions, but I'll try to help.

A 3x3 determinant can be used to obtain the roots of a system of 3 simultaneous linear equations.

The determinant can be obtained by solving a general 3x3 system by algebraic means.

The question about quadratics: the fundamental theorem of algebra guarantees 2 roots of a quadratic equation...and you can find those roots by solving the general quadratic by completing the square which gives you the quadratic formula.

Answer 2

The determinant can either be positive, negative, or 0. If its positive, then it means that your equation has 2 distinct real solutions. If it's negative, it will have 2 imaginary solutions. And if it's 0, it means there is a multiplicity, or 2 real solutions that are the same. In terms of the graph of this quadratic, a positive determinant means the graph of the parabola crosses the x-axis twice. If the determinant is negative, the parabola doesn't cross the x-axis. If the determinant is 0, it means the vertex of the parabola is on the x-axis. The determinant comes from calculating b² - 4(a)(c) in the quadratic formula. You know that every quadratic has two roots because the formula includes the (+) symbol, which means that you go through the formula twice: once by adding the square root of the determinant, and one by subtracting the square root of the determinant.