# If a2 + 1 = a, then the value of a12 + a6 + 1 is

**Answer: (2) 3
Explanation:-
**a

^{2}+ 1 = a

Both side divide by a,

a

^{2}/a + 1/a = a/a

a + 1/a + 1

On squaring of both sides,

(a + 1/a)

^{2}= 1

^{2}

a

^{2}+ 1/a

^{2}+ 2 x a x 1/a = 1

a

^{2} + 1/a

^{2}+ 2 = 1

a

^{2}+ 1/a

^{2}= 1 – 2 = -1 …………………….. (1)

On cubing of both sides,

(a

^{2}+ 1/a

^{2})

^{3}= ( – 1)

^{3}

(a

^{2})

^{3}+ (1/a

^{2})

^{3 }+ 3 x a

^{2}x 1/a

^{2}(a

^{2}+ 1/a

^{2}) = (-1) x (-1) x (- 1)

a

^{6}+ 1/a

^{6}+ 3(a

^{2}+ 1/a

^{2}) = – 1

a

^{6}+ 1/a

^{6}+ 3 (-1) = – 1

a

^{6}+ 1/a

^{6}+ 1 = 3 …………………….. [From equation (1)]

(a

^{12}+ 1 + a

^{6})/a

^{6}= 3

a

^{12}+ a

^{6}+ 1 = 3

Hence, the answer is (2) 3.

(a^12+1+a^6)/a^6=3

a^12+1+a^6=3*a^6

how it became 3 sir please explain……….

**a ^{2} +1 = a**

**a ^{2} –a +1 =0**

**multiply by (a+1) on both side**

**(a+1)( a ^{2} –a +1) =0**

**a ^{3} + 1^{3} = 0 ( Since a^{3} + 1^{3} = (a+1)( a^{2} –a +1)**

**a ^{3} = -1**

**Now put the value of a ^{3} in given expression**

**a ^{12} + a^{6} +1= (-1)^{4} + (-1)^{2} + 1 =1+1+1 = 3**

**a ^{2} +1 = a**

**a ^{2} –a +1 =0**

**multiply by (a+1) on both side**

**(a+1)( a ^{2} –a +1) =0**

**a ^{3} + 1^{3} = 0 ( Since a^{3} + 1^{3} = (a+1)( a^{2} –a +1)**

**a ^{3} = -1**

**Now put the value of a ^{3} in given expression**

**a ^{12} + a^{6} +1= (-1)^{4} + (-1)^{2} + 1 =1+1+1 = 3**