Question

If `x=3+sqrt8`, what is the value of `x^2+1/x^2`?

Answer 1

See a solution process below:

Explanation:

To find the value of the expression in the problem we need to substitute `color(red)(3 + sqrt(8))` for each occurrence of `color(red)(x)` in the expression and then simplify the resulting expression:

`color(red)(x)^2 + 1/color(red)(x)^2` become:

`color(red)((3 + sqrt(8)))^2 + 1/color(red)((3 + sqrt(8)))^2`

We can use this rule to square the terms:

`(a + b)^2 = a^2 + 2ab + b^2`

Substituting `3` for `a` and substituting `sqrt(8)` for `b` gives:

`(3 + sqrt(8))^2 => 3^2 + (2 * 3 * sqrt(8)) + sqrt(8)^2 =>`

`9 + 6sqrt(8) + 8 =>`

`17 + 6sqrt(8)`

We can substitute this for both occurrences of `color(red)((3 + sqrt(8)))^2` giving:

`17 + 6sqrt(8) + 1/(17 + 6sqrt(8))`

We can put the term on the left over a common denominator so we can add the fractions giving:

`((17 + 6sqrt(8))/(17 + 6sqrt(8))) xx (17 + 6sqrt(8)) + 1/(17 + 6sqrt(8)) =>`

`(17 + 6sqrt(8))^2/(17 + 6sqrt(8)) + 1/(17 + 6sqrt(8))`

We can square the numerator of the fraction on the left using the same rule giving:

`(17^2 + (2 * 17 * 6sqrt(8)) + (6sqrt(8))^2)/(17 + 6sqrt(8)) + 1/(17 + 6sqrt(8)) =>`

`(289 + 2046sqrt(8) + (36 * 8))/(17 + 6sqrt(8)) + 1/(17 + 6sqrt(8)) =>`

`(289 + 2046sqrt(8) + 288))/(17 + 6sqrt(8)) + 1/(17 + 6sqrt(8)) =>`

`(577 + 2046sqrt(8))/(17 + 6sqrt(8)) + 1/(17 + 6sqrt(8)) =>`

`(577 + 2046sqrt(8) + 1)/(17 + 6sqrt(8)) =>`

`(578 + 2046sqrt(8))/(17 + 6sqrt(8))`

Answer 2

`frac(577 + 408 sqrt(2))(17 + 12 sqrt(2))`

Explanation:

We have: `x = 3 + sqrt(8)`

Squaring both sides of the equation:

`Rightarrow x^(2) = (3 + sqrt(8))^(2)`

`Rightarrow x^(2) = (3)^(2) + 2 cdot 3 cdot sqrt(8) + (sqrt(8))^(2)`

`Rightarrow x^(2) = 9 + 6 sqrt(8) + 8`

`Rightarrow x^(2) = 17 + 6 cdot 2 sqrt(2)`

`Rightarrow x^(2) = 17 + 12 sqrt(2)`

`therefore x^(2) + frac(1)(x^(2)) = 17 + 12 sqrt(2) + frac(1)(17 + 12 sqrt(2))`

`Rightarrow x^(2) + frac(1)(x^(2)) = frac((17 + 12 sqrt(2))^(2) + 1)(17 + 12 sqrt(2))`

`Rightarrow x^(2) + frac(1)(x^(2)) = frac((17)^(2) + 2 cdot 17 cdot 12 sqrt(2) + (12 sqrt(2))^(2) + 1)(17 + 12 sqrt(2))`

`Rightarrow x^(2) + frac(1)(x^(2)) = frac(289 + 408 sqrt(2) + 288)(17 + 12 sqrt(2))`

`Rightarrow x^(2) + frac(1)(x^(2)) = frac(577 + 408 sqrt(2))(17 + 12 sqrt(2))`