Solving Rational Equations.

Question

Solve $1+\frac{1}{x^2}=\frac{3}{x}$

I tried to cross multiply after combining 1 + $\frac{1}{x^2}$ but it just came out weird. How do I solve this?

Answer 1

$1+\frac 1{x^2}=\frac 3x\rightarrow x^2-3x+1=0$

$x=\frac{3\pm\sqrt{9-4}}{2}$

$x=\frac{3\pm\sqrt 5}2$

Answer 2

Multiply by $x^2$ throughout

$$x^2\left(1+\dfrac{1}{x^2}=\dfrac{3}{x}\right)$$

Gives $x^2+1=3x\tag{1}$

Arrange $(1)$ into general quadratic form $$ax+bx+c=0\tag{2}$$

$$x^2-3x+1=0\tag{3}$$

From $(3)$

$a=1,b=-3,c=1$

Recall $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\tag{4}$$

Substitute values of $a,b,c$ into $(4)$

Gives $x=\dfrac{3\pm\sqrt{(-3)^2-4(1)(1)}}{2(1)}$

$x=\dfrac{3+\sqrt{5}}{2}$ and $x=\dfrac{3-\sqrt{5}}{2}$ are solutions