Question

What subsitution should be used to rewrite x^8-3x^4+2=0 as a quadratic equation​

Answer 1

Replace x with
`y^(1/4)`

Explanation:

The general form of a quadratic equation is
`ax^(2)` + bx + c.

To find a substitution we have to think of a way of substituting ‘x’ to make it have a power of 2,

Here, we will use
`y^(1/4)`

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

let x =
`y^(1/4)`

so it gives

(
`y^(1/4)`)^8 – 3(
`y^(1/4)` )^4 +2 = 0

`y^(2)` – 3y + 2 = 0

Solving the equation

`y^(2)` - 2y – y + 2 =0

(
`y^(2)` – 2y) -1(y – 2) = 0

y(y - 2) -1(y - 2) = 0

(y-1) (y-2)

:- y = 1 or y = 2

Since x =
`y^(1/4)`

When y = 1

X =
`1^(1/4)`

X =
`4√(1)`

X = 1

and when y = 2

X =
`2^(1/4)`

X =
`4√(2)`

X = 1.1892...

Answer 2

Replace x^2 with m

Explanation:

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

We want to write this as a quadratic, which means as am^2+bm +c

x^8 needs to become m^2

Let m = x^4

Then m^2 = x^4^2 = x^8

Our equation becomes

m^2 -3m +2 =0

Replace x^2 with m