Grade what is the units digit of the simplest form of ? Answer: Solution

Ans.
Solution:
Divide 2x² – x - 2 = 0 by 2x gives = .
Square both sides of the equation, we get =
= and squaring both sides again will yield =
Another squaring of both sides of the equation will results to
=

11. 
    If (a + b)⁴ + (a – b)⁴ = 4177 and a² – b² = 24, where a and b are both real numbers, find the value of (a² + b²).
    

Ans. 36 ½
Solution:
(a - b)(a + b) = 24, which may results to a – b = 3 and a + b = 8
By checking, (a + b)⁴ + (a – b)⁴ = 4177, 8⁴ + 3⁴ = 4177, hence adding squaring a – b = 3 and a + b = 8 gives
a² – 2ab + b² = 9 and a² + 2ab + b² = 64.
Adding these equations and dividing both sides by 2, we get
a² + b² = or 36 ½

12. 
    Given that the minimum value of f(x) = 3x² – 12x + 19 is F and the maximum value of g(x) = -4x² – 40x – 115 is G, find the value of F – G?
    

Ans. 22
Solution:
f(x) = 3x² – 12x + 19 in vertex form is f(x) = 3(x – 2)² + 7 with F = 7
g(x) = -4x² – 40x – 115 in vertex form is g(x) = -4(x + 5)² – 15 with G = - 15

F – G = 7 – (-15) = 22.

13. 
    The equation (3 – x)³ + (4 – x)³ = (7 – 2x)³ has roots x₁, x₂, and x₃. What is the value of (x₁ + x₂ + x₃)?