Answer:
The value of m is [tex]\frac{-5+\sqrt{34}}{6} \text { or } \frac{-5-\sqrt{34}}{6}[/tex] by using quadratic formula
Solution:
Given, expression is [tex]m+\frac{2}{3}=\frac{1}{4 m}-1[/tex]
Now, we have to solve the above given expression.
[tex]\text { Now, } \mathrm{m}+\frac{2}{3}=\frac{1}{4 m}-1[/tex]
By multiplying the equation with m, we get
[tex]\begin{array}{l}{m^{2}+\frac{2}{3} m+m=\frac{1}{4}} \\\\ {m^{2}+m\left(\frac{2}{3}+1\right)=\frac{1}{4}} \\\\ {m^{2}+\frac{5}{3} m=\frac{1}{4}}\end{array}[/tex]
[tex]\begin{array}{l}{12 m^{2}+20 m=3} \\ {12 m^{2}+20 m-3=0}\end{array}[/tex]
Now, let us use quadratic formula
[tex]\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex]
Here in our problem, a = 12, b = 20, c = -3
[tex]\begin{array}{l}{m=\frac{-20 \pm \sqrt{20^{2}-4 \times 12 \times(-3)}}{2 \times 12}} \\\\ {=\frac{-20 \pm \sqrt{400+144}}{24}} \\\\ {=\frac{-20 \pm \sqrt{544}}{24}} \\\\ {=\frac{-20 \pm 4 \sqrt{34}}{24}=\frac{-5 \pm \sqrt{34}}{6}} \\\\ {=\frac{-5+\sqrt{34}}{6} \text { or } \frac{-5-\sqrt{34}}{6}}\end{array}[/tex]
Hence the value of m is [tex]\frac{-5+\sqrt{34}}{6} \text { or } \frac{-5-\sqrt{34}}{6}[/tex] by using quadratic formula
