calculate one expression that depends on other expressionwe know 3x=1-x^2 and calculate x^4+6x^3+12x^2+9x-4

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to calculate x^4+6x^3+12x^2+9x-4 and we know that 3x=1-x^2

x^4+6x^3+12x^2+9x-4

=> x^2^2 + 6*x*x^2 + 12x^2 + 9x - 4

=> (1 - 3x)^2 +6*x*(1 - 3x) + 12x^2 + 9x - 4

=> 1 + 9x^2 - 6x + 6x - 18x^2 + 12x^2 + 9x - 4

=> 1 + 3x^2 + 9x - 4

=> 1 + 3x^2 + 3 - 3x^2 - 4

=> 0

Therefore x^4+6x^3+12x^2+9x-4 = 0

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll re-write the constraint from enunciation:

x^2+3x-1=0 gives x^2+3x = 1

We'll raise to square both sides:

(x^2 + 3x)^2 = 1^2

We'll expand the square:

x^4 + 6x^3 + 9x^2 = 1 (1)

We'll re-write the expresison that has to be calculated, with respect to (1):

x^4 + 6x^3 + 9x^2 + 3x^2 + 9x - 4

We'll group the first 3 terms:

(x^4 + 6x^3 + 9x^2) + 3x^2 + 9x - 4

We'll substitute x^4 + 6x^3 + 9x^2 by it's value 1:

1 + 3x^2 + 9x - 4

We'll re-write -4 = -3-1

1 + 3x^2 + 9x - 3 - 1

We'll eliminate like terms and we'll get:

3x^2 + 9x - 3

We'll factorize by 3:

3(x^2+3x-1)

But x^2+3x-1 = 0 is the constraint from enunciation, so the expresison to be calculated will cancel out.

x^4+6x^3+12x^2+9x-4 = 0

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