Eric Nicholas K's answer to tiffany's Secondary 1 Maths Singapore question.
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Luna, since you have tagged lots of supposedly Sec 2 Express or Sec 3 Express questions as Sec 1 Maths, I presume are either in the IP stream or you are learning this way in advance on your own.
I have used a Sec 2 Express or Sec 3 Express approach to do this, so let me know if you need more explanation.
I have used a Sec 2 Express or Sec 3 Express approach to do this, so let me know if you need more explanation.
Date Posted:
4 years ago
Alternative :
x² = 5x - 1
x² + 1 = 5x
(x² + 1)² = (5x)²
x⁴ + 2x² + 1 = 25x²
x⁴ + 1 = 23x²
(x⁴ + 1)² = (23x²)²
x^8 + 2x⁴ + 1 = 529x⁴
x^8 + 1 = 527x⁴
2x^8 + 2 = 1054x⁴
2x⁴ + 2/x⁴ = 1054
(dividing whole equation by 4)
x² = 5x - 1
x² + 1 = 5x
(x² + 1)² = (5x)²
x⁴ + 2x² + 1 = 25x²
x⁴ + 1 = 23x²
(x⁴ + 1)² = (23x²)²
x^8 + 2x⁴ + 1 = 529x⁴
x^8 + 1 = 527x⁴
2x^8 + 2 = 1054x⁴
2x⁴ + 2/x⁴ = 1054
(dividing whole equation by 4)
Or
x² = 5x - 1 ①
2x⁴ + 2/x⁴
= 2(x²)² + 4(x²)(1/x²) + 2(1/x²)² - 4(x²)(1/x²)
= 2(x² + 1/x²)² - 4
= 2(x² + 2x(1/x) + (1/x)² - 2x(1/x))² - 4
= 2( (x + 1/x)² - 2)² - 4
= 2( ((x² + 1)/x)² - 2)² - 4
= 2( ((5x - 1 + 1)/x)² - 2)² - 4 from ①
= 2 (5² - 2)² - 4
= 2(23)² - 4
= 1058 - 4 = 1054
x² = 5x - 1 ①
2x⁴ + 2/x⁴
= 2(x²)² + 4(x²)(1/x²) + 2(1/x²)² - 4(x²)(1/x²)
= 2(x² + 1/x²)² - 4
= 2(x² + 2x(1/x) + (1/x)² - 2x(1/x))² - 4
= 2( (x + 1/x)² - 2)² - 4
= 2( ((x² + 1)/x)² - 2)² - 4
= 2( ((5x - 1 + 1)/x)² - 2)² - 4 from ①
= 2 (5² - 2)² - 4
= 2(23)² - 4
= 1058 - 4 = 1054
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