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Unformatted text preview: Certificate Mathematics in Action Full Solutions 4A 4 More about Equations • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Follow-up Exercise p. 162 1. By substituting x 2 = u into the equation x 4 – 10 x 2 + 9 = 0, we have u 2 – 10 u + 9 = 0 ( u – 1)( u – 9) = 0 u = 1 or u = 9 ∵ x 2 = u ∴ x 2 = 1 or x 2 = 9 x = ± 1 or x = ± 3 ∴ The real roots of the equation are –3, –1, 1 and 3. 2. By substituting x 2 = u into the equation x 4 + 3 x 2 – 4 = 0, we have u 2 + 3 u – 4 = 0 ( u – 1)( u + 4) = 0 u = 1 or u = – 4 ∵ x 2 = u ∴ x 2 = 1 or x 2 = –4 (rejected) x = ± 1 ∴ The real roots of the equation are –1 and 1. 3. By substituting x 2 = u into the equation 4 x 4 – 17 x 2 + 4 = 0, we have 4 u 2 – 17 u + 4 = 0 (4 u – 1)( u – 4) = 0 u = 4 1 or u = 4 ∵ x 2 = u ∴ x 2 = 4 1 or x 2 = 4 x = 2 1 ± or x = ± 2 ∴ The real roots of the equation are –2, 2 1- , 2 1 and 2. 4. By substituting x 3 = u into the equation x 6 + 2 x 3 + 1 = 0, we have u 2 + 2 u + 1 = 0 ( u + 1) 2 = 0 u = –1 ∵ x 3 = u ∴ x 3 = –1 x = –1 ∴ The real root of the equation is –1. p. 163 1. 8 4 2-- x x = 2 2 2 8 4 -- x x = 2 2 x 2 – 4 x – 8 = 4 x 2 – 4 x – 12 = 0 ( x + 2)( x – 6) = 0 x = –2 or x = 6 Checking: When x = –2, 8 4 2-- x x = 8 ) 2 ( 4 ) 2 ( 2-- ×-- = 2 When x = 6, 8 4 2-- x x = 8 6 4 6 2- ×- = 2 ∴ The real roots of the equation are –2 and 6. 2. x + x 3 = 4 x – 4 = – x 3 ( x – 4) 2 = ( 29 2 3 x- x 2 – 8 x + 16 = 9 x x 2 – 17 x + 16 = 0 ( x – 1)( x – 16) = 0 x = 1 or x = 16 Checking: When x = 1, x + x 3 = 1 + 1 3 = 4 When x = 16, x + x 3 = 16 + 16 3 = 28 ≠ 4 ∴ The real root of the equation is 1. Alternative Solution By substituting x = u into the equation x + x 3 = 4, we have u 2 + 3 u = 4 u 2 + 3 u – 4 = 0 ( u – 1)( u + 4) = 0 u = 1 or u = –4 ∵ x = u ∴ x = 1 or x = –4 (rejected) x = 1 ∴ The real root of the equation is 1. 3. x – 2 3- x = 0 x = 2 3- x x 2 = ( 29 2 2 3- x x 2 = 9 x – 18 x 2 – 9 x + 18 = 0 ( x – 3)( x – 6) = 0 x = 3 or x = 6 Checking: When x = 3, x – 2 3- x = 3 – 2 3 3- = 0 When x = 6, x – 2 3- x = 6 – 2 6 3- = 0 ∴ The real roots of the equation are 3 and 6. 4. x + 2 2- x = 10 2 2- x = 10 – x ( 29 2 2 2- x = (10 – x ) 2 4 x – 8 = 100 – 20 x + x 2 x 2 – 24 x + 108 = 0 ( x – 6)( x – 18) = 0 x = 6 or x = 18 Checking: 83 4 More about Equations When x = 6, x + 2 2- x = 6 + 2 6 2- = 10 When x = 18, x + 2 2- x = 18 + 2 18 2- = 26 ≠ 10 ∴ The real root of the equation is 6. p.168 For questions 1 to 4, refer to the graph below: 1. y = 3 x – 5 x 1 2 3 y –2 1 4 ∵ The two graphs intersect at only one point (2, 1)....
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