If x^{4} = x² + 1, where x > 0, then what is 2x^{4} equal to?

Correct Answer is : 3+√53 + √53+√5

Correct Answer is : 3+√53 + √53+√5

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UPSC CDS II 2025 Mathematics Paper

Question : 13 of 100

Marks: +1, -0

x^{4} = x^{2} + 1

x > 0

, then what is 2x^{4} equal to?

Solution:

Concept:

Use substitution to simplify a quartic equation into a quadratic, then solve step by step.

Explanation:

Let

y = x^{2}

. Then

x^{4} = y^{2}

. The given equation

x^{4} = x^{2} + 1

becomes

y^{2} = y + 1

Rearrange:

y^{2} - y - 1 = 0

. Solve using the quadratic formula:

y = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}

Since

x > 0

x^{2} > 0

, so take the positive root:

y = \frac{1 + \sqrt{5}}{2}

. Thus

x^{2} = \frac{1 + \sqrt{5}}{2}

Now

x^{4} = y^{2} = \left(\frac{1 + \sqrt{5}}{2}\right)^{2} = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2}

Multiply both sides by 2:

2x^{4} = 3 + \sqrt{5}

Answer:

3 + \sqrt{5}

(Option B)

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