What substitution should be used to rewrite x⁸ - 3x⁴ + 2 = 0 as a quadratic equation?u = x²u = x⁴u = x⁸u = x¹⁶

Solution:

The given quadratic equation is

x⁸ - 3x⁴ + 2 = 0

Let us substitute u = x⁴

So the quadratic equation becomes

u² - 3u + 2 = 0

The standard form of the quadratic equation ax² + bx + c = 0 and the formula used is

x = [-b ± √(b² - 4ac)]/ 2a

Here a = 1, b = -3 and c = 2

Substituting these values in the formula

x = [-(-3) ± √((-3)² - 4 × 1 × 2)]/ (2 × 1)

By further simplification

x = [3 ± √(9 - 8)]/ 2

x = [3 ± √1]/ 2

So we get

x = (3 + 1)/2 = 4/2 = 2

x = (3 - 1)/2 = 2/2 = 1

We could easily find the value of x when we do this u-substitution, where u = x⁴. Now x = ± 1, ±∜2

Therefore, u = x⁴ should be substituted to rewrite as a quadratic equation.

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What substitution should be used to rewrite x⁸ - 3x⁴ + 2 = 0 as a quadratic equation?u = x²u = x⁴u = x⁸u = x¹⁶

Summary:

The substitution u = x⁴ should be used to rewrite x⁸ - 3x⁴ + 2 = 0 as a quadratic equation.