Respuesta :
[tex]-12x^2 - 25x + 5 + x^{3} = 0[/tex]
[tex]x^{3} - 12x^{2} - 25x + 5 = 0[/tex]
[tex]x = \sqrt[3]{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d}{2a}) + \sqrt{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d}{2a})^{2} + (\frac{c}{3a} - \frac{b^{2}}{9a^{2}})^{3}}} + \sqrt[3]{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d}{2a}) - \sqrt{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d}{2a})^{2} + (\frac{c}{3a} - \frac{b^{2}}{9a^{2}})^{3}}} - \frac{b}{3a}[/tex]
[tex]x = \sqrt[3]{(\frac{-(-12)^{3}}{27(1)^{3}} + \frac{(-12)(-25)}{6(1)^{2}} - \frac{5}{2(1)}) + \sqrt{(\frac{-(-12)^{3}}{27(1)^{3}} + \frac{(-12)(-25)}{6(1)^{2}} - \frac{5}{2(1)})^{2} + (\frac{-25}{3(1)} - \frac{-(-25)^{2}}{9(1)^{2}})^{3}}} + \sqrt[3]{(\frac{-(-12)^{3}}{27(1)^{3}}}} + \frac{(-12)(-25)}{6(1)^{2}} - \frac{5}{2(1)}) - \sqrt{(\frac{-(-12)^{3}}{27(1)^{3}} + \frac{(-12)(-25)}{6(1)^{2}} - \frac{5}{2(1)})^{2} + (\frac{-25}{3(1)} - \frac{-(-25)^{2}}{9(1)^{2}})^{3} - \frac{-12}{3(1)}}}[/tex]
[tex]x = \sqrt[3]{(\frac{-(-1728)}{27(1)} + \frac{300}{6(1)} - \frac{5}{2}) + \sqrt{(\frac{-(-1728)}{27(1)^{3}} + \frac{300}{6(1)} - \frac{5}{2})^{2} + (\frac{-25}{3(1)} - \frac{144}{9(1)})^{3}}}} + \sqrt[3]{(\frac{-(-1728)}{27(1)} + \frac{300}{6(1)} - \frac{5}{2}) - \sqrt{(\frac{-(-1728)}{27(1)^{3}} + \frac{300}{6(1)} - \frac{5}{2})^{2} + (\frac{-25}{3(1)} - \frac{144}{9(1)})^{3}}}} - \frac{-12}{3}[/tex]
[tex]x = \sqrt[3]{(\frac{1728}{27} + \frac{300}{6} - 2\frac{1}{2}) + \sqrt{(\frac{1728}{27} + \frac{300}{6} - 2\frac{1}{2})^{2} + (\frac{-25}{3} - \frac{144}{9})^{3}}} + \sqrt[3]{(\frac{1728}{27} + \frac{300}{6} - 2\frac{1}{2}) - \sqrt{(\frac{1728}{27} + \frac{300}{6} - 2\frac{1}{2})^{2} + (\frac{-25}{3} - \frac{144}{9})^{3}}} - 4[/tex]
[tex]x = \sqrt[3]{(64 + 50 - 2\frac{1}{2}) + \sqrt{(64 + 50 - 2\frac{1}{2})^{2} + (-8\frac{1}{3} - 16)^{3}}} + sqrt[3]{(64 + 50 - 2\frac{1}{2}) - \sqrt{(64 + 50 - 2\frac{1}{2})^{2} + (-8\frac{1}{3} - 16)^{3}}} - 4[/tex]
[tex]x = \sqrt[3]{(114 - 2\frac{1}{2}) + \sqrt{(114 - 2\frac{1}{2})^{2} + (-24\frac{1}{3})^{3}}} + \sqrt[3]{(114 - 2\frac{1}{2}) - \sqrt{(114 - 2\frac{1}{2})^{2} + (-24\frac{1}{3})^{3}}} - 4[/tex]
[tex]x = \sqrt[3]{(112\frac{1}{2}) + \sqrt{(112\frac{1}{2})^{2} - (24\frac{1}{3})^{3}}} - \sqrt[3]{(112\frac{1}{2}) + \sqrt{(112\frac{1}{2})^{2} - (24\frac{1}{3})^{3}}} - 4[/tex]
[tex]x = \sqrt[3]{112\frac{1}{2} + \sqrt{12656.25 - 14408.037}} + \sqrt[3]{112\frac{1}{2} + \sqrt{12656.25 - 14408.037}} - 4[/tex]
[tex]x = \sqrt[3]{112\frac{1}{2} + \sqrt{-1751.787}} + \sqrt[3]{112\frac{1}{2} - \sqrt{-1751.787}} - 4[/tex]
[tex]x = \sqrt[3]{112\frac{1}{2} + 41.855i} + \sqrt[3]{112\frac{1}{2} - 41.855i} - 4[/tex]
[tex]x = -4 + \sqrt[3]{112\frac{1}{2} + 41.855i} + \sqrt[3]{112\frac{1}{2} - 41.855i}[/tex]
[tex]x^{3} - 12x^{2} - 25x + 5 = 0[/tex]
[tex]x = \sqrt[3]{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d}{2a}) + \sqrt{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d}{2a})^{2} + (\frac{c}{3a} - \frac{b^{2}}{9a^{2}})^{3}}} + \sqrt[3]{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d}{2a}) - \sqrt{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d}{2a})^{2} + (\frac{c}{3a} - \frac{b^{2}}{9a^{2}})^{3}}} - \frac{b}{3a}[/tex]
[tex]x = \sqrt[3]{(\frac{-(-12)^{3}}{27(1)^{3}} + \frac{(-12)(-25)}{6(1)^{2}} - \frac{5}{2(1)}) + \sqrt{(\frac{-(-12)^{3}}{27(1)^{3}} + \frac{(-12)(-25)}{6(1)^{2}} - \frac{5}{2(1)})^{2} + (\frac{-25}{3(1)} - \frac{-(-25)^{2}}{9(1)^{2}})^{3}}} + \sqrt[3]{(\frac{-(-12)^{3}}{27(1)^{3}}}} + \frac{(-12)(-25)}{6(1)^{2}} - \frac{5}{2(1)}) - \sqrt{(\frac{-(-12)^{3}}{27(1)^{3}} + \frac{(-12)(-25)}{6(1)^{2}} - \frac{5}{2(1)})^{2} + (\frac{-25}{3(1)} - \frac{-(-25)^{2}}{9(1)^{2}})^{3} - \frac{-12}{3(1)}}}[/tex]
[tex]x = \sqrt[3]{(\frac{-(-1728)}{27(1)} + \frac{300}{6(1)} - \frac{5}{2}) + \sqrt{(\frac{-(-1728)}{27(1)^{3}} + \frac{300}{6(1)} - \frac{5}{2})^{2} + (\frac{-25}{3(1)} - \frac{144}{9(1)})^{3}}}} + \sqrt[3]{(\frac{-(-1728)}{27(1)} + \frac{300}{6(1)} - \frac{5}{2}) - \sqrt{(\frac{-(-1728)}{27(1)^{3}} + \frac{300}{6(1)} - \frac{5}{2})^{2} + (\frac{-25}{3(1)} - \frac{144}{9(1)})^{3}}}} - \frac{-12}{3}[/tex]
[tex]x = \sqrt[3]{(\frac{1728}{27} + \frac{300}{6} - 2\frac{1}{2}) + \sqrt{(\frac{1728}{27} + \frac{300}{6} - 2\frac{1}{2})^{2} + (\frac{-25}{3} - \frac{144}{9})^{3}}} + \sqrt[3]{(\frac{1728}{27} + \frac{300}{6} - 2\frac{1}{2}) - \sqrt{(\frac{1728}{27} + \frac{300}{6} - 2\frac{1}{2})^{2} + (\frac{-25}{3} - \frac{144}{9})^{3}}} - 4[/tex]
[tex]x = \sqrt[3]{(64 + 50 - 2\frac{1}{2}) + \sqrt{(64 + 50 - 2\frac{1}{2})^{2} + (-8\frac{1}{3} - 16)^{3}}} + sqrt[3]{(64 + 50 - 2\frac{1}{2}) - \sqrt{(64 + 50 - 2\frac{1}{2})^{2} + (-8\frac{1}{3} - 16)^{3}}} - 4[/tex]
[tex]x = \sqrt[3]{(114 - 2\frac{1}{2}) + \sqrt{(114 - 2\frac{1}{2})^{2} + (-24\frac{1}{3})^{3}}} + \sqrt[3]{(114 - 2\frac{1}{2}) - \sqrt{(114 - 2\frac{1}{2})^{2} + (-24\frac{1}{3})^{3}}} - 4[/tex]
[tex]x = \sqrt[3]{(112\frac{1}{2}) + \sqrt{(112\frac{1}{2})^{2} - (24\frac{1}{3})^{3}}} - \sqrt[3]{(112\frac{1}{2}) + \sqrt{(112\frac{1}{2})^{2} - (24\frac{1}{3})^{3}}} - 4[/tex]
[tex]x = \sqrt[3]{112\frac{1}{2} + \sqrt{12656.25 - 14408.037}} + \sqrt[3]{112\frac{1}{2} + \sqrt{12656.25 - 14408.037}} - 4[/tex]
[tex]x = \sqrt[3]{112\frac{1}{2} + \sqrt{-1751.787}} + \sqrt[3]{112\frac{1}{2} - \sqrt{-1751.787}} - 4[/tex]
[tex]x = \sqrt[3]{112\frac{1}{2} + 41.855i} + \sqrt[3]{112\frac{1}{2} - 41.855i} - 4[/tex]
[tex]x = -4 + \sqrt[3]{112\frac{1}{2} + 41.855i} + \sqrt[3]{112\frac{1}{2} - 41.855i}[/tex]
Answer:
There are 3 roots of the given equation.
Step-by-step explanation:
Given the equation
[tex]-12x^2-25x+5+x^3=0[/tex]
we have to tell the number of roots of the given equation.
As the number of roots for an equation is equal to degree.
The degree of a polynomial is the highest power of its monomials with non-zero coefficients.
Hence, number of roots is the highest power in the equation.
Now, the equation is [tex]-12x^2-25x+5+x^3=0[/tex]
The highest power i.e degree of equation is 3.
hence, there are 3 roots of the given equation.
