Question

find the discriminant of the following equation to determine the number and type of solutions it will have?

Answer 1

(a) The equation $2h^{2} +7h+4=0$ has two irrational solutions.

(b) The equation $m^{2} =-40m-400$ has one rational solution.

(c) The equation $14r^{2} =5-7r$ has two irrational solutions.

(d) The equation $7w^{2} -w=-9$ has two imaginary solutions.

(e) The equation $3f-9f^{2} =6$ has two imaginary solutions.

Explanation:

(a) Solving the equation $2h^{2} +7h+4=0$ , we get the solutions,

$h=\frac{-7+\sqrt{17}}{4}$ and $h=\frac{-7-\sqrt{17}}{4}$

Thus, $h=-0.719223$ and $h=-2.78077$

Hence, the equation $2h^{2} +7h+4=0$ has two irrational solutions.

(b) Solving the equation $m^{2} =-40m-400$ , we get the solution,

$m=-20$

Hence, the equation $m^{2} =-40m-400$ has one rational solution.

(c) Solving the equation $14r^{2} =5-7r$ , we get the solutions,

$r=\frac{-7+\sqrt{329}}{28}$ and $r=\frac{-7-\sqrt{329}}{28}$

Thus, $r=-0.39779$ and $r=-0.89779$

Hence, the equation $14r^{2} =5-7r$ has two irrational solutions.

(d) Solving the equation $7w^{2} -w=-9$ , we get the solutions,

$w=\frac{1}{14}+i \frac{\sqrt{251}}{14}$ and $w=\frac{1}{14}-i \frac{\sqrt{251}}{14}$

Hence, the equation $7w^{2} -w=-9$ has two imaginary solutions.

(e)  Solving the equation $3f-9f^{2} =6$ , we get the solutions,

$f=\frac{1}{6}-i \frac{\sqrt{23}}{6}$ and $f=\frac{1}{6}+i \frac{\sqrt{23}}{6}$

Hence, the equation $3f-9f^{2} =6$ has two imaginary solutions.