Question

X^-2+4x^-1+3=0 solve by making appropriate substitution

Answer 1

`x = - 1 \: or \: x = - (1)/( 3)`

Step-by-step explanation

The given equation is:

`{x}^( - 2) + 4 {x}^( - 1) + 3 = 0`

Recall that:

`{a}^( - m) = \frac{1}{ {a}^(m) }`

`\frac{1}{ {x}^(2) } + (4)/(x) + 3 = 0`

Or

`{( (1)/(x) )}^(2) + 4( (1)/(x) ) + 3 = 0`

Let

`u = (1)/(x)`

Our equation then becomes:

`{u}^(2) + 4u + 3 = 0`

The factors of 3 that add up to 4 are:

`{u}^(2) + 3u + u + 3`

`u(u + 3) + 1(u + 3) = 0`

`(u + 1)(u + 3) = 0`

`u + 1 = 0 \: or \: u + 3 = 0`

`u = - 1 \: or \: u = - 3`

This implies that:

`(1)/(x) = - 1 \: or \: (1)/(x) = - 3`

`x = - 1 \: or \: x = - (1)/( 3)`