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how-to-show-f-x-x-4-2x-3-5x-2-16x-20-in-the-form-of-x-2-x-a-2-4-x-b-2-

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Question Number 162351 by ZiYangLee last updated on 29/Dec/21
how to show f(x)=x^4 +2x^3 +5x^2 −16x−20 in the form of (x^2 +x+a)^2 −4(x+b)^2 .
howtoshowf(x)=x4+2x3+5x2−16x−20intheformof(x2+x+a)2−4(x+b)2.
Answered by Rasheed.Sindhi last updated on 29/Dec/21
f(x)=x^4 +2x^3 +5x^2 −16x−20 g(x)=(x^2 +x+a)^2 −4(x+b)^2 (x^2 +x+a)^2 −4(x+b)^2 =x^4 +x^2 +a^2 +2x^3 +2ax+2ax^2 −4x^2 −8bx−b^2 =x^4 +2x^3 +2ax^2 −4x^2 +x^2 +a^2 +2ax−8bx−4b^2 =x^4 +2x^3 +(2a−3)x^2 +(2a−8b)x+a^2 −4b^2 Comparing coefficients of f & g 2a−3=5 ∧ 2a−8b=−16 ∧ a^2 −4b^2 =−20 a=4 ∧b=3 ∧ 4^2 −4.3^2 =−20✓ g(x)=(x^2 +x+4)^2 −4(x+3)^2
f(x)=x4+2x3+5x2−16x−20g(x)=(x2+x+a)2−4(x+b)2(x2+x+a)2−4(x+b)2=x4+x2+a2+2x3+2ax+2ax2−4x2−8bx−b2=x4+2x3+2ax2−4x2+x2+a2+2ax−8bx−4b2=x4+2x3+(2a−3)x2+(2a−8b)x+a2−4b2Comparingcoefficientsoff&g2a−3=5∧2a−8b=−16∧a2−4b2=−20a=4∧b=3∧42−4.32=−20✓g(x)=(x2+x+4)2−4(x+3)2
Commented by mr W last updated on 29/Dec/21
nice working!
niceworking!
Commented by Rasheed.Sindhi last updated on 29/Dec/21
ThanX for mentioning my mistake Sir!
ThanXformentioningmymistakeSir!
Commented by Rasheed.Sindhi last updated on 29/Dec/21
Than_(sir!) ^(Lo_(f) t) X
ThanLoftsir!X

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