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¶ … Value of Money
Quadratic Formula
Info: When using the quadratic formula to solve a quadratic equation ax2 + bx + c = 0, the discriminant is b2-4ac. This discriminant can be positive, zero, or negative. (When the discriminate is negative, then we have the square root of a negative number. This is called an imaginary number, sqrt (-1) = i.)
Explain what the value of the discriminant means to the graph of y = ax2 + bx + c. Hint: Chose values of a, b and c to create a particular discriminant. Then, graph the corresponding equation.
Search the Cybrary and Internet. In the real world, where might these imaginary numbers be used?
The discriminant b2-4ac is used to identify three possible solution cases for quadratic equations: one real solution, two real solutions, and an imaginary solution. For the first case (one real solution), it will show a parabola that touches the x-axis at a single point. For example, the equation y = x^2-2x + 1 will produce the following graph:
For this case, the discriminant is b^2-4ac = -2^2-4*1*1 = 4-4 = 0. The parabola touches the x-axis at a single point (2, 0), which is the solution if y = 0. For the second case (two real solutions), it will show a parabola that touches the x-axis at two points. For example, the equation y = x^2-3x + 2 will have the following graph:
In this case, the discriminant is b^2-4ac = -3^2-4*1*2 = 9-8 = 1. The parabola touches the x-axis at points (1, 0) and (2, 0), which are the two solutions when y = 0. For the third case (imaginary solution), it won't show a graph in the Cartesian plane unless we include the complex plane. For example, the equation y = x^2-4x + 13 will have the discriminant b^2-4ac = -4^2-4*1*13 = 16-52 = -36. In the real world, imaginary numbers are used in various fields like electrical engineering (signal analysis), quantum mechanics, physics (theory of relativity), fractals, and fluid dynamics.
Bibliography:
Complex number. (2006, May 19). In Wikipedia, The Free Encyclopedia. Retrieved 07:19, May 22, 2006, at http://en.wikipedia.org/w/index.php?title=Complex_number&oldid=53969038.
Quadratic equation. (2006, May 21). In Wikipedia, The Free Encyclopedia. Retrieved 07:19, May 22, 2006, at http://en.wikipedia.org/w/index.php?title=Quadratic_equation&oldid=54349372.
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