74-hw1s - MATH 74 HOMEWORK 1 1(a). If x satisfies the given...

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Unformatted text preview: MATH 74 HOMEWORK 1 1(a). If x satisfies the given equation, then x 2 satisfies a quadratic equation (namely ( x 2 ) 2- 6( x 2 ) + 7 = 0), so using the quadratic formula we see that x 2 = 6 p (- 6) 2- 4 1 7 2 1 = 6 8 2 = 6 4 2 2 = 6 2 2 2 = 3 2 , that is, either x 2 = 3+ 2 or x 2 = 3- 2. Since both of these numbers are positive, they each have two real square roots. Conclusion: if x satisfies x 4- 6 x 2 + 7 = 0, then x is one of the four numbers p 3 + 2,- p 3 + 2, p 3- 2,- p 3- 2, and any one of these is possible (as you can check directly, or take for granted as part of the quadratic formula). 1(b). Note that x 2 satisfies a quadratic equation, ( x 2 ) 2- 2( x 2 )- 1 = 0, so by the quadratic formula x 2 = 2 p 4- 4 1 (- 1) 2 = 2 8 2 = 2 4 2 2 = 2 2 2 2 = 1 2 , that is, either x 2 = 1 + 2 or x 2 = 1- 2. Because 1- 2 is negative , however, we know this second case does not happen (the square of any real number is non-...
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74-hw1s - MATH 74 HOMEWORK 1 1(a). If x satisfies the given...

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