Find all values of p, such that the roots of f ( x ) = x 3 </msup> + 8

enrotlavaec

enrotlavaec

Answered question

2022-06-06

Find all values of p, such that the roots of f ( x ) = x 3 + 8 x 2 + p x + 12 satisfy x 1 + x 2 = x 3

Answer & Explanation

kpgt1z

kpgt1z

Beginner2022-06-07Added 23 answers

Since x 1 + x 2 + x 3 = 8 and x 1 + x 2 = x 3 , then x 3 = 4. Then
f ( 4 ) = ( 4 ) 3 + 8 ( 4 ) 2 + p ( 4 ) + 12 = 76 4 p = 0
Then p = 19
Brenden Tran

Brenden Tran

Beginner2022-06-08Added 9 answers

Apart from the other answer, if you want to use Vieta's formulas, you can see that x 1 x 2 = 12 4 = 3. Using this in x 1 x 2 + x 1 x 3 + x 2 x 3 = p, we get
x 1 x 2 + ( x 1 + x 2 ) x 3 = p x 1 x 2 + x 3 2 = p 3 + 16 = p
So, the answer is p = 19

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