find all real solutions of the equation. 1/x^3+ 8/x^2 + 16/x=0

Question

1# Find all real solution of the equation.  (If there is no real solution, enter no real solution).  Show all work square root 126-10x = 2x

#2. Find all real solution of the equation. If no solution enter no real solution.  Show all work.  x^5-81x=0

#3 2x^2+9x+1=0 solve by completing the square and show all work.

#4. Find all real solution of the equation.  Show all work if no real solution please state that.  35x^2+12X-36=0

#5. Use the discriminant to determine the number of real solution of the equation.  Do not solve the equation.  x^2-8x+1=0

#6. Arectangular garden is 2 ft longer than it is wide.  It area is 783 ft2.  What are its dimensions?  show all work

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Answer 1

1. √(126-10x)=2x, 126-10x=4x², 4x²+10x-126=0, 2x²+5x-63=0=(2x-9)(x+7), x=9/2 or -7. Substitute back in original: √(126-45)=√81=9 is satisfied; √(126+70)=√196=14 is not satisfied, because 2x=-14. So the solution is x=9/2.
2. x⁵-81x=x(x⁴-81)=x(x²-9)(x²+9)=x(x-3)(x+3)(x²+9)=0; solution is x=0, 3, -3.
3. 2x²+9x+1=0; x²+9x/2+1/2=0; (x²+9x/2+81/16)-81/16+1/2=0; (x+9/4)²-(81-8)/16=0; ((x+9/4)+√73/4)((x+9/4)-√73/4)=0; solution is x=-9/4-√73/4=-4.386, -9/4+√73/4=-0.114 (approx).
4. 35x²+12x-36=0=(7x-6)(5x+6), so x=6/7, -6/5.
5. Discriminant=8²-4×1=64-4=60. √60 is used to find the solutions.
6. If w=width, length=w+2; w(w+2)=783, w²+2w=783, w²+2w-783=0=(x+29)(x-27), width=27ft so length=29ft.