ডেইলি কুইজ [২০০ দিন]

a² + b² = 89
বা, 2(a² + b²) = 178
বা, (a + b)² + (a - b)² = 178
বা, (a - b)² = 178 - (a + b)²
বা, (a - b)² = 178 - (13)²
বা, (a - b)² = 178 - 169 = 9
∴ a - b = 3

(x + 1/x)² = (x - 1/x)² + 4.x.1/x
= 5² + 4
(x + 1/x)² = 29
∴ x + 1/x = √29

∴ x² - 1/x²
= (x + 1/x)(x - 1/x)
= √29 × 5
= 5√29

2x - 2/x = 1
বা, 2(x-1/x) = 1
বা, x - 1/x = 1/2

∴ (x - 1/x)² = 1/4
বা, x² + 1/x² - 2 = 1/4
বা, x² + 1/x² = 1/4 + 2 = 9/4

এখন, (x² + 1/x²)² = (9/4)²
⇒ x⁴ + 1/x⁴ + 2 = 81/16
⇒ x⁴ + 1/x⁴ = 81/16 - 2
∴ x⁴ + 1/x⁴ = 49/16

x + 1/x = √3
বা, (x + 1/x)³ = (√3)²
বা, x³ + 1/x³ + 3.x.1/x(x + 1/x) = 3√3
বা, x³ +1/x³ + 3√3 = 3√3
বা, x³ + 1/x³ = 3√3 - 3√3
∴ x³ + 1/x³ = 0

9x² - ax + 1
= (3x)² + 6x + 1
= (3x)² + 2.3x.1 + 1²
= (3x + 1)²
∴ a = -6 হলে সংখ্যাটি পূর্ণ বর্গ হবে।

x² - 8x - 8y + 16 + y²
= x² + y² + 16 - 2.x.4 - 2.y.4 - 2xy
= (x + y - 4)² - 2xy

⁴√x³ = 2
বা, x³ = 2⁴
বা, x³ = 16
∴ x⁶ = 16² 
= 256

2{(x²)² + (y²)²}
= (x² + y²)² + (x² - y²)²
= 5² + 3²
= 34
∴ 2(x⁴ + y⁴) = 34
বা, x⁴ + y⁴ = 34/2
= 17

1/a + 1/b
= (a + b)/ab
= √((a + b)²)/ab
= √(a² + b² + 2ab)/ab
= √{288 + (2 × 18)}/18
= √324/18
= 18/18
= 1

x = 7 + 4√3
∴ 1/x = 7 - 4√3
x + 1/x
= 7 + 4√3 + 7 - 4√3
= 14
(x + 1/x)² = 14²
বা, x² + 1/x² + 2.x.1/x = 196
বা, x² + 1/x² = 196 - 2
= 194

x² - y² + 4y - 4
= x² - (y² - 4y + 4)
= x² - (y - 2)²
= (x + y - 2)(x - y + 2)

9x² + 9x - 4
= 9x² + 12x - 3x - 4
= 3x(3x + 4) - 1(3x + 4)
= (3x + 4)(3x - 1)

x⁶ - 1
= (x³)² - 1
= (x³ + 1)(x³ - 1)
= (x + 1)(x² - x + 1)(x + 1)(x² + x + 1)

x + 6, f(x) = x² - kx - 48 এর একটি উৎপাদক
∴ f(-6) = 0
বা, (-6)² - k(-6) - 48 = 0
বা, 36 + 6k - 48 = 0
বা, 6k = 12
∴ k = 2

x⁴ + 4
= (x²)² + 2²
= (x² + 2)² - 2.x².2
= (x² + 2)² - (2x)²
= (x² + 2x + 2)(x² - 2x +2)

y² + 2xy - 2x - 1
= y² - 1 + 2xy - 2x
= (y + 1)(y - 1) + 2x(y - 1)
= (y - 1)(2x + y + 1)

x = 3 হলে সমীকরণটি সিদ্ধ হয়
∴ সমাধান x = 3

(x + 1)(x - 1) = x² - 1
∴ x² - 5)x² - 1(1
             x² - 5
             -------
                   4
∴ ভাগশেষ = 4

16 - 2x³ = 0
বা, -2(x³ - 8) = 0
বা, x³ - 8 = 0
বা, (x - 2)(x² + 2x + 4) = 0

হয়, x - 2 = 0
∴ x = 2
অথবা, x² + 2x + 4 = 0