(a) The equation [tex]2h^{2} +7h+4=0[/tex] has two irrational solutions.
(b) The equation [tex]m^{2} =-40m-400[/tex] has one rational solution.
(c) The equation [tex]14r^{2} =5-7r[/tex] has two irrational solutions.
(d) The equation [tex]7w^{2} -w=-9[/tex] has two imaginary solutions.
(e) The equation [tex]3f-9f^{2} =6[/tex] has two imaginary solutions.
Explanation:
(a) Solving the equation [tex]2h^{2} +7h+4=0[/tex] , we get the solutions,
[tex]h=\frac{-7+\sqrt{17}}{4}[/tex] and [tex]h=\frac{-7-\sqrt{17}}{4}[/tex]
Thus, [tex]h=-0.719223[/tex] and [tex]h=-2.78077[/tex]
Hence, the equation [tex]2h^{2} +7h+4=0[/tex] has two irrational solutions.
(b) Solving the equation [tex]m^{2} =-40m-400[/tex] , we get the solution,
[tex]m=-20[/tex]
Hence, the equation [tex]m^{2} =-40m-400[/tex] has one rational solution.
(c) Solving the equation [tex]14r^{2} =5-7r[/tex] , we get the solutions,
[tex]r=\frac{-7+\sqrt{329}}{28}[/tex] and [tex]r=\frac{-7-\sqrt{329}}{28}[/tex]
Thus, [tex]r=-0.39779[/tex] and [tex]r=-0.89779[/tex]
Hence, the equation [tex]14r^{2} =5-7r[/tex] has two irrational solutions.
(d) Solving the equation [tex]7w^{2} -w=-9[/tex] , we get the solutions,
[tex]w=\frac{1}{14}+i \frac{\sqrt{251}}{14}[/tex] and [tex]w=\frac{1}{14}-i \frac{\sqrt{251}}{14}[/tex]
Hence, the equation [tex]7w^{2} -w=-9[/tex] has two imaginary solutions.
(e) Solving the equation [tex]3f-9f^{2} =6[/tex] , we get the solutions,
[tex]f=\frac{1}{6}-i \frac{\sqrt{23}}{6}[/tex] and [tex]f=\frac{1}{6}+i \frac{\sqrt{23}}{6}[/tex]
Hence, the equation [tex]3f-9f^{2} =6[/tex] has two imaginary solutions.
