Question

What substitution should be used to rewrite x8 – 3x4 + 2 = 0 as a quadratic equation?

Answer 1

substitution should be p = x⁴

Explanation:

It is given that,

x⁸ - 3x⁴ + 2 = 0

we can rewrite the equation,

(x⁴)² - 3x⁴ + 2 =0

To find the substitution

Here we can see that x⁴ is common in two terms of the given equation

we can substitute p instead of x⁴, the equation becomes,

p² - 3p +2 = 0

Therefore substitution should be used to rewrite x8 – 3x4 + 2 = 0 as a quadratic equation is p = x⁴

Answer 2

The substitution is

`u = x ^ 4`

`u ^ 2 -3u +2 = 0`

Explanation:

We have the 8th degree polynomial equation

`x ^ 8 - 3x^4 -+2 = 0`

To rewrite the equation as a quadratic function, take the common factor of the term x with the smallest exponent, in this case it is
`x ^ 4`.

Now make a change of variable

`u = x ^ 4`.

So rewriting the equation in terms of u, we have:

`u ^ 2 -3u +2 = 0`

Now the initial equation became a quadratic equation

Factoring is left:

`(u-2) (u-1) = 0`

`u = 2` and
`u = 1`

`x ^ 4 = 2` and
`x ^ 4 = 1`