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IF a,b,c are real numbers and 3b^2-8ac

IF a,b,c are real numbers and 3b^2-8ac<0,then the equation ax^4+bx^3+cx^2+5x+7=0 cannot have all roots real. prove this.

Grade:12

1 Answers

mycroft holmes
272 Points
12 years ago

The wording of the question is a dead giveaway.

 

Suppose all the roots r1, r2, r3 and r4 are real, then we must have

 

so that

 

Now, from Vieta's formulas, . So, if all roots are real, we must necessarily have  

 

or .

 

Since we are given that  it implies that all roots cannot be real.

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