74-hw1s

# 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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