Explanation Given Information: The provided polynomial equation is x 4 + x 2 − 1 = 0 . Formula used: The value of x in the equation a x 2 + b x + c = 0 can be given by the quadratic formula , x = − b ± b 2 − 4 a c 2 a where a , b and c are real numbers with a ≠ 0 . Calculation: Consider the polynomial equation x 4 + x 2 − 1 = 0 . The above polynomial equation can be written as ( x 2 ) 2 + x 2 − 1 = 0 Let x 2 = t put in the polynomial equation ( x 2 ) 2 + x 2 − 1 = 0 t 2 + t − 1 = 0 Compare the above equation with the equation a x 2 + b x + c = 0 , a = 1 , b = 1 and c = − 1 Put the values of a , b , c in the quadratic formula x = − b ± b 2 − 4 a c 2 a . t = − 1 ± ( 1 ) 2 − 4 ⋅ 1 ( − 1 ) 2 ⋅ 1 = − 1 ± 1 + 4 2 = − 1 ± 5 2 Further simplify, Put − 1 ± 5 2 in x 2 = t , x 2 = t = − 1 ± 5 2 Take square root from both sides, x = ± − 1 ± 5 2 Check: For x = − 1 ± 5 2 Substitute x = − 1 ± 5 2 in the equation x 4 + x 2 − 1 = 0 then simplify, ( − 1 ± 5 2 ) 4 + ( − 1 ± 5 2 ) 2 − 1 = ? 0 ( − 1 ± 5 2 ) 2 + ( − 1 ± 5 2 ) − 1 = ? 0 Taking positive ( + ) sign from the above equation then simplify, ( − 1 + 5 2 ) 2 + ( − 1 + 5 2 ) − 1 = ? 0 1 + 5 − 2 5 4 + − 1 + 5 2 − 1 = ? 0 6 − 2 5 − 2 + 2 5 − 4 4 = ? 0 0 = 0 Which is true

7th Edition

ISBN: 9781337514309

Author: Waner

Publisher: Cengage

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Chapter 0.5, Problem 29E

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To calculate: The all real solutions in the polynomial equation x4+x2−1=0.

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Chapter 0.5, Problem 28E

Chapter 0.5, Problem 30E

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Chapter 0 Solutions

Applied Calculus

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Ch. 0.1 - Prob. 11E Ch. 0.1 - Prob. 12E Ch. 0.1 - Prob. 13E Ch. 0.1 - Calculate each expression in Exercises 124, giving...Ch. 0.1 - Prob. 15E Ch. 0.1 - Prob. 16E Ch. 0.1 - Calculate each expression in Exercises 124, giving...Ch. 0.1 - Prob. 18E Ch. 0.1 - Prob. 19E Ch. 0.1 - Prob. 20E Ch. 0.1 - Prob. 21E Ch. 0.1 - Prob. 22E Ch. 0.1 - Prob. 23E Ch. 0.1 - Prob. 24E Ch. 0.1 - Prob. 25E Ch. 0.1 - Prob. 26E Ch. 0.1 - Prob. 27E Ch. 0.1 - Prob. 28E Ch. 0.1 - Prob. 29E Ch. 0.1 - Prob. 30E Ch. 0.1 - Prob. 31E Ch. 0.1 - Prob. 32E Ch. 0.1 - Prob. 33E Ch. 0.1 - Prob. 34E Ch. 0.1 - Prob. 35E Ch. 0.1 - Prob. 36E Ch. 0.1 - Convert each expression in Exercises 25-50 into...Ch. 0.1 - Prob. 38E Ch. 0.1 - Prob. 39E Ch. 0.1 - Prob. 40E Ch. 0.1 - Prob. 41E Ch. 0.1 - Prob. 42E Ch. 0.1 - Prob. 43E Ch. 0.1 - Prob. 44E Ch. 0.1 - Prob. 45E Ch. 0.1 - Prob. 46E Ch. 0.1 - Prob. 47E Ch. 0.1 - Prob. 48E Ch. 0.1 - Prob. 49E Ch. 0.1 - Prob. 50E Ch. 0.2 - Evaluate the expression sin Exercises 116. 33 Ch. 0.2 - Prob. 2E Ch. 0.2 - Evaluate the expression sin Exercises 116. 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In Exercises 2946,...Ch. 0.8 - Prob. 32E Ch. 0.8 - Let a=log2,b=log3, and c=log7. In Exercises 2946,...Ch. 0.8 - Prob. 34E Ch. 0.8 - Prob. 35E Ch. 0.8 - Prob. 36E Ch. 0.8 - Prob. 37E Ch. 0.8 - Prob. 38E Ch. 0.8 - Prob. 39E Ch. 0.8 - Prob. 40E Ch. 0.8 - Prob. 41E Ch. 0.8 - Prob. 42E Ch. 0.8 - Prob. 43E Ch. 0.8 - Prob. 44E Ch. 0.8 - Prob. 45E Ch. 0.8 - Prob. 46E Ch. 0.8 - Prob. 47E Ch. 0.8 - Prob. 48E Ch. 0.8 - Prob. 49E Ch. 0.8 - Prob. 50E Ch. 0.8 - Prob. 51E Ch. 0.8 - Prob. 52E Ch. 0.8 - Prob. 53E Ch. 0.8 - Prob. 54E Ch. 0.8 - Prob. 55E Ch. 0.8 - Prob. 56E

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The all real solutions in the polynomial equation x 4 + x 2 − 1 = 0 .

Solution Summary: The author calculates the polynomial equation x4+2-1=0 using a quadratic formula.

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Chapter 0 Solutions