- #1
Bruno Tolentino
- 97
- 0
A quadratic equation in this format [tex] x² - 2 A x + B² = 0 [/tex]
The roots x1 and x2 are therefore: [tex]x_1 = x_1(u,v) = u + v[/tex] [tex]x_2 = x_2(u,v) = u - v[/tex] Or: [tex]x_1 = x_1(a, b) = \frac{a+b}{2} + \frac{|a-b|}{2}[/tex] [tex]x_2 = x_2(a, b) = \frac{a+b}{2} - \frac{|a-b|}{2}[/tex]
Until here, these relationships are known. No problem! The problem begins when I ask: "how are these relationships for the cubic equation?"
I know that analog cubic equation is [tex] x^3 - 3 A x^2 + 3 B^2 x - C^3 = 0 [/tex] and that
[tex] A = \frac{a+b+c}{3} [/tex] [tex] B = \sqrt[2]{\frac{ca + ab + bc}{3}} [/tex] [tex] C = \sqrt[3]{abc} [/tex]
a, b and c are the roots. The roots can be called too of x1, x2 and x3. But, by analogy with the relationships above, how I can to express x1, x2 and x3 in terms of a linear combination of a, b and c?
[tex]x_1 = x_1(a, b, c) = ? [/tex] [tex]x_2 = x_2(a, b, c) = ? [/tex] [tex]x_3 = x_3(a, b, c) = ? [/tex]
And how express x1, x2 and x3 in terms of u, v and w?
[tex]x_1 = x_1(u, v, w) = ? [/tex] [tex]x_2 = x_2(u, v, w) = ? [/tex] [tex]x_3 = x_3(u, v, w) = ? [/tex]
can be modified and expressed like: x² - 2 (u) x + (u² - v²) = 0.where:
A = (a + b)×(1/2)
B = (a × b)^(1/2)
a and b are the roots and the roots can be expressed in terms of midpoint and radius
u = midpoint
v = radius
a = u + v
b = u - v
The roots x1 and x2 are therefore: [tex]x_1 = x_1(u,v) = u + v[/tex] [tex]x_2 = x_2(u,v) = u - v[/tex] Or: [tex]x_1 = x_1(a, b) = \frac{a+b}{2} + \frac{|a-b|}{2}[/tex] [tex]x_2 = x_2(a, b) = \frac{a+b}{2} - \frac{|a-b|}{2}[/tex]
Until here, these relationships are known. No problem! The problem begins when I ask: "how are these relationships for the cubic equation?"
I know that analog cubic equation is [tex] x^3 - 3 A x^2 + 3 B^2 x - C^3 = 0 [/tex] and that
[tex] A = \frac{a+b+c}{3} [/tex] [tex] B = \sqrt[2]{\frac{ca + ab + bc}{3}} [/tex] [tex] C = \sqrt[3]{abc} [/tex]
a, b and c are the roots. The roots can be called too of x1, x2 and x3. But, by analogy with the relationships above, how I can to express x1, x2 and x3 in terms of a linear combination of a, b and c?
[tex]x_1 = x_1(a, b, c) = ? [/tex] [tex]x_2 = x_2(a, b, c) = ? [/tex] [tex]x_3 = x_3(a, b, c) = ? [/tex]
And how express x1, x2 and x3 in terms of u, v and w?
[tex]x_1 = x_1(u, v, w) = ? [/tex] [tex]x_2 = x_2(u, v, w) = ? [/tex] [tex]x_3 = x_3(u, v, w) = ? [/tex]