given a^2+3a-1=0, calculate a^4+6a^3+12a^2+9a-4?
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Given a^2 + 3a - 1 = 0, the result for a^4 + 6a^3 + 12a^2 + 9a - 4 is required.
a^4 + 6a^3 + 12a^2 + 9a - 4
Let's split the terms and arrange them so that we can arrive at a^2 + 3a - 1 for every three terms by eliminating common factors.
=> a^4 + 3a^3 - a^2 + 3a^3 + 9a^2 - 3a + 3a^2 + 9a - 3 + a^2 + 3a - 1
=> a^2(a^2 + 3a - 1) + 3a( a^2 + 3a - 1) + 3(a^2 + 3a - 1) + (a^2 + 3a - 1)
=> 0
The required value of a^4 + 6a^3 + 12a^2 + 9a - 4 = 0
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We'll re-write the constraint from enunciation:
a^2+3a-1=0 gives a^2+3a = 1
We'll raise to square both sides:
(a^2 + 3a)^2 = 1^2
We'll expand the square:
a^4 + 6a^3 + 9a^2 = 1 (1)
We'll re-write the expresison that has to be calculated, with respect to (1):
a^4 + 6a^3 + 9a^2 + 3a^2 + 9a - 4
We'll group the first 3 terms:
(a^4 + 6a^3 + 9a^2) + 3a^2 + 9a - 4
1 + 3a^2 + 9a - 4
We'll re-write -4 = -3-1
1 + 3a^2 + 9a - 3 - 1
We'll eliminate like terms and we'll get:
3a^2 + 9a - 3
We'll factorize by 3:
3(a^2 + 3a - 1)
But, from enunciation a^2 + 3a - 1 = 0, so the expresison to be calculated will be zero.
a^4+6a^3+12a^2+9a-4 = 0
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