Question

If x² – 6x + 1 = 0, then find the value of x¹⁰ + x⁸ – 1154x⁶ – 1154x⁴

1. 4x
2. -4x
3. -6x
4. 3x
5. 5x

Answer 1

Correct Answer - Option 3 : -6x

Given:

x² – 6x + 1 = 0

Concept Used:

Divide the given equation by x to get equation in the form of (x) + (1/x)

Formula Used:

(x) + (1/x) = k , then (x²) + (1/x²) = k² – 2

(x²) + (1/x²) = m, then (x⁴) + (1/x⁴) = m² - 2

Calculation:

(x) + (1/x) = 6

Then,

(x²) + (1/x²) = 6² - 2

⇒ (x²) + (1/x²) = 36 - 2

⇒ (x²) + (1/x²) = 34

Now,

(x⁴) + (1/x⁴) = 34² - 2

⇒ (x⁴) + (1/x⁴) = 1156 - 2

⇒ (x⁴) + (1/x⁴) = 1154

Now substitute the value of 1154 in given equation

K = x¹⁰ + x⁸ – 1154x⁶ – 1154x⁴

⇒ K = x¹⁰ + x⁸ – [(x⁴) + (1/x⁴)] x⁶ – [(x⁴) + (1/x⁴)]x⁴

⇒ K = x¹⁰ + x⁸ – [x¹⁰ + x²] – [x⁸ + 1]

⇒ K = x¹⁰ + x⁸ – x¹⁰ - x² – x⁸ - 1

⇒ K = -(x² + 1)

Substituting the value of (x² + 1) from given equation

⇒ K = -(6x)

∴ The value of x¹⁰ + x⁸ – 1154x⁶ – 1154x⁴ is -6x