GATE 2018 IN GA Q.9: solution and answer

GATE 2018 · INGA Q.9 · 2 marks · MCQ

If $x^{2} + x - 1 = 0$ what is the value of $x^{4} + \frac{1}{x ^{4}}$ ?

(A)
1
(B)
5
(C)
7
(D)
9

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Answer and solution

ANSWER

$x^{4} + \frac{1}{x ^{4}} = 7$ . The answer is option (C).

Official answer: CSource: Official answer key, GATE 2018 IN (IIT Guwahati)

$x = 0$ does not satisfy the equation (it would give $- 1 = 0$ ), so you can divide every term by $x$ : $x + 1 - \frac{1}{x} = 0$ , which gives $x - \frac{1}{x} = - 1$ .
Square both sides using $(a - b)^{2} = a^{2} - 2 ab + b^{2}$ . The cross term is $2 \cdot x \cdot \frac{1}{x} = 2$ : $x^{2} - 2 + \frac{1}{x ^{2}} = 1 \Rightarrow x^{2} + \frac{1}{x ^{2}} = 3$
Square again using $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ . The cross term is $2 \cdot x^{2} \cdot \frac{1}{x ^{2}} = 2$ : $x^{4} + 2 + \frac{1}{x ^{4}} = 9 \Rightarrow x^{4} + \frac{1}{x ^{4}} = 7$
Check with an actual root: $x = \frac{- 1 + 5}{2} \approx 0.618$ gives $x^{4} \approx 0.1459$ and $\frac{1}{x ^{4}} \approx 6.854$ , which add up to $\approx 7.000$ . The other root, $x = \frac{- 1 - 5}{2} \approx - 1.618$ , is the negative reciprocal of the first, so it gives the same two numbers the other way round and the same sum.
So $x^{4} + \frac{1}{x ^{4}} = 7$ , which is option (C).

KEY IDEA

Divide the quadratic by $x$ to get $x - \frac{1}{x}$ , then square twice. Each squaring turns $x^{n} \pm \frac{1}{x ^{n}}$ into $x^{2 n} + \frac{1}{x ^{2 n}}$ plus or minus 2, because $x^{n} \cdot \frac{1}{x ^{n}} = 1$ . You never have to solve for $x$ .

WATCH OUT FOR

Getting the sign of $x - \frac{1}{x}$ wrong. Dividing by $x$ gives $x + 1 - \frac{1}{x} = 0$ , so $x - \frac{1}{x} = - 1$ , not $x + \frac{1}{x} = - 1$ . If you take $x + \frac{1}{x} = - 1$ , the first squaring gives $x^{2} + \frac{1}{x ^{2}} = 1 - 2 = - 1$ , which is impossible for real $x$ and leads to $x^{4} + \frac{1}{x ^{4}} = - 1$ , not any of the options. A second slip is forgetting the $\pm 2$ cross term when squaring, which gives 1 or 9 (options A and D) instead of 7.

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GATE 2018 IN GA Q.9: solution and answer

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