Question

Solve:
x + 1/x = 4 1/4


Ans: 4,1/4​

Answer 1

`\boxed{x = (1)/(4)}` and
`\boxed{x = 4}`

Explanation:

Given equation:

`x + (1)/(x) = 4(1)/(4)`

Step-1: Convert the mixed fraction on the R.H.S into improper fraction

`x + (1)/(x) = 4(1)/(4)`

`x + (1)/(x) = (4 *4 + 1)/(4)`

`x + (1)/(x) = (16 + 1)/(4)`

`x + (1)/(x) = (17)/(4)`

Step-2: Make common denominators on the L.H.S:

`x + (1)/(x) = (17)/(4)`

`(x^(2) )/(x) + (1)/(x) = (17)/(4)`

Step-3: Combine the denominators on the L.H.S

`(x^(2) )/(x) + (1)/(x) = (17)/(4)`

`(x^(2) +1)/(x) = (17)/(4)`

Step-4: Use cross multiplication

`(x^(2) +1)/(x) = (17)/(4)`

`x^(2) +1} = (17x)/(4)`

`4(x^(2) +1}) = {17x}`

Step-5: Simplify the distributive property

`4(x^(2) +1}) = {17x}`

`4x^(2) +4} = {17x}`

`-17x + 4x^(2) +4} = 0`

Step-6: Change "-17x" to "-16x - x" as it is equivalent

`-17x + 4x^(2) +4} = 0`

`(-16x - x) + 4x^(2) +4} = 0`

Step-7: Factor the common terms

`(-16x - x) + 4x^(2) +4} = 0`

`-16x - x + 4x^(2) +4} = 0`

`4x(-4 + x) - 1(x - 4) = 0`

Step-8: Group the terms

`4x(-4 + x) - 1(x - 4) = 0`

`(x - 4)(4x - 1) = 0`

Step-9i: Use cross multiplication for (x - 4)

`(x - 4)(4x - 1) = 0`

`x - 4 = (0)/(4x - 1 ) = 0`

Step-9ii: Use cross multiplication for (4x - 1)

`(x - 4)(4x - 1) = 0`

`4x - 1 = (0)/(x - 4) = 0`

Thus
`x - 4 = 0` and
`4x - 1 = 0`.

Step-10: Simplify both equations

`4x - 1 = 0`
`x - 4 = 0`

`4x = 0 + 1`
`x = 0 + 4`

`4x = 1`
`\boxed{x = 4}`

`\boxed{x = (1)/(4)}`

Answer 2

x = 4, 1/4

solving steps

`\sf \rightarrow x + (1)/(x)=4(1)/(4)`

make the denominators same

`\sf \rightarrow (x(x))/(x) + (1)/(x)=4(1)/(4)`

simplify the following

`\sf \rightarrow (x^2)/(x) + (1)/(x)=(17)/(4)`

join both fractions together

`\sf \rightarrow (x^2+1)/(x)=(17)/(4)`

cross multiply

`\sf \rightarrow 4(x^2+1)=17(x)`

simplify

`\sf \rightarrow 4x^2-17(x)+4=0`

completing square

`\sf \rightarrow 4x^2-16(x)-x+4=0`

factor

`\sf \rightarrow 4x(x-4)-1(x-4)=0`

group the variables

`\sf \rightarrow (4x-1)(x-4)=0`

simplify

`\sf \rightarrow (x-4)=0, \ (4x-1) =0`

final answer

`\sf \rightarrow x=4, \ x =(1)/(4)`