If for non-zero, x, x2 – 4x – 1 = 0, the value of x2

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x² – 4x – 1 = 0
⇒ x² –1 = 4x
Dividing by x

x −	1	= 4
	x

On squaring both sides,

	x −	1		²	= 16
		x

⇒ x² +	1	– 2 = 16
	x²

⇒ x² +	1	= 16 + 2 = 18
	x²

Second Method :
Using Rule 5,
Here, x² – 4x – 1 = 0
⇒ x² – 1 = 4 x

⇒ x² –	1	= 4
	x

We know that

x² +	1	=		x −	1		²	+ 2
	x²				x

= 4² + 2 = 18