What substitution should be used to rewrite 4x¹² - 5x⁶ - 14 = 0 as a quadratic equation?
u = x^2, u = x³, u = x⁶, u = x¹²

Solution:

The given quadratic equation is

4x¹² - 5x⁶ - 14 = 0

Let us assume u = x⁶

4u² - 5u - 14 = 0

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

x = [-b ± √(b² - 4ac)]/ 2a

Here a = 4, b = -5 and c = -14

Substituting these values in the formula

x = [-(-5) ± √((-5)² - 4 × 4 × -14)]/ (2 × 4)

By further simplification

x = [5 ± √(25 + 224)]/ 8

x = [5 ± √249]/ 8

Therefore, u = x⁶ should be used to rewrite as a quadratic equation.

What substitution should be used to rewrite 4x¹² - 5x⁶ - 14 = 0 as a quadratic equation?
u = x^2, u = x³, u = x⁶, u = x¹²

Summary:

The substitution u = x⁶ should be used to rewrite 4x¹² - 5x⁶ - 14 = 0 as a quadratic equation.