Respuesta :
The value of x is [tex]x=\frac{-1+\sqrt{37}}{12}[/tex] and [tex]x=\frac{-1-\sqrt{37}}{12}[/tex]
Step-by-step explanation:
The equation is [tex]\frac{1}{x}-\frac{2}{3}=4 x[/tex]
Subtracting by [tex]4x[/tex] on both sides,
[tex]\frac{1}{x}-\frac{2}{3}-4 x=0[/tex]
Taking LCM,
[tex]\frac{3-12 x^{2}-2 x}{3 x}=0[/tex]
Multiplying by 3x on both sides,
[tex]-12 x^{2}-2 x+3=0[/tex]
Dividing by (-) on both sides,
[tex]12 x^{2}+2 x-3=0[/tex]
Using quadratic formula, we can solve for x.
[tex]\begin{aligned}x &=\frac{-2 \pm \sqrt{2^{2}-4 \cdot 12 \cdot(-3)}}{2 \cdot 12} \\&=\frac{-2 \pm \sqrt{4+144}}{2 \cdot 144} \\&=\frac{-2 \pm \sqrt{148}}{24} \\&=\frac{-2 \pm 2 \sqrt{37}}{24}\end{aligned}[/tex]
Taking out common term 2, we get,
[tex]\begin{array}{l}{x=\frac{-2(1 \pm \sqrt{37})}{24}} \\{x=\frac{-1 \pm \sqrt{37}}{12}}\end{array}[/tex]
Thus, the value of x is [tex]x=\frac{-1+\sqrt{37}}{12}[/tex] and [tex]x=\frac{-1-\sqrt{37}}{12}[/tex]
