You do in fact get a negative number so the solutions are not real but imaginary. Good!
Answer:
1.) ![x=8\sqrt{2}](https://tex.z-dn.net/?f=x%3D8%5Csqrt%7B2%7D)
2.) ![x=14](https://tex.z-dn.net/?f=x%3D14)
3.) ![x=18\sqrt{2}](https://tex.z-dn.net/?f=x%3D18%5Csqrt%7B2%7D)
4.) ![x=9\sqrt{2}](https://tex.z-dn.net/?f=x%3D9%5Csqrt%7B2%7D)
5.) ![x=4\sqrt{2}](https://tex.z-dn.net/?f=x%3D4%5Csqrt%7B2%7D)
6.) ![x=5\sqrt{2}](https://tex.z-dn.net/?f=x%3D5%5Csqrt%7B2%7D)
7.) ![x=12\sqrt{2}](https://tex.z-dn.net/?f=x%3D12%5Csqrt%7B2%7D)
Step-by-step explanation:
Use the 45°-45°-90° formula:
![hypotenuse=\sqrt{2}*leg](https://tex.z-dn.net/?f=hypotenuse%3D%5Csqrt%7B2%7D%2Aleg)
1.) Insert values:
![x=\sqrt{2}*8](https://tex.z-dn.net/?f=x%3D%5Csqrt%7B2%7D%2A8)
Simplify:
![x=8\sqrt{2}](https://tex.z-dn.net/?f=x%3D8%5Csqrt%7B2%7D)
2.) In a 45°-45°-90° angle, the legs have the same value.
![x=14](https://tex.z-dn.net/?f=x%3D14)
3.) x is the hypotenuse. Insert values:
![x=\sqrt{2}*18](https://tex.z-dn.net/?f=x%3D%5Csqrt%7B2%7D%2A18)
Simplify:
![x=18\sqrt{2}](https://tex.z-dn.net/?f=x%3D18%5Csqrt%7B2%7D)
4.) Insert values:
![18=\sqrt{2}*x](https://tex.z-dn.net/?f=18%3D%5Csqrt%7B2%7D%2Ax)
Divide
from both sides:
![\frac{18}{\sqrt{2}}=\frac{\sqrt{2}*x}{\sqrt{2}} \\\\\frac{18}{\sqrt{2}}=x](https://tex.z-dn.net/?f=%5Cfrac%7B18%7D%7B%5Csqrt%7B2%7D%7D%3D%5Cfrac%7B%5Csqrt%7B2%7D%2Ax%7D%7B%5Csqrt%7B2%7D%7D%20%20%5C%5C%5C%5C%5Cfrac%7B18%7D%7B%5Csqrt%7B2%7D%7D%3Dx)
Rationalize the left side:
![\frac{\sqrt{2}}{\sqrt{2}}*\frac{18}{\sqrt{2}}=x\\\\\frac{18\sqrt{2}}{\sqrt{4}}\\\\\frac{18\sqrt{2}}{2} \\\\9\sqrt{2}=x](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B2%7D%7D%2A%5Cfrac%7B18%7D%7B%5Csqrt%7B2%7D%7D%3Dx%5C%5C%5C%5C%5Cfrac%7B18%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B4%7D%7D%5C%5C%5C%5C%5Cfrac%7B18%5Csqrt%7B2%7D%7D%7B2%7D%20%5C%5C%5C%5C9%5Csqrt%7B2%7D%3Dx)
Simplify:
![x=9\sqrt{2}](https://tex.z-dn.net/?f=x%3D9%5Csqrt%7B2%7D)
5.) Insert values:
![8=\sqrt{2}*x](https://tex.z-dn.net/?f=8%3D%5Csqrt%7B2%7D%2Ax)
Divide
from both sides and rationalize:
![\frac{8}{\sqrt{2}}=\frac{\sqrt{2}*x }{\sqrt{2}}\\\\\frac{8}{\sqrt{2}}=x](https://tex.z-dn.net/?f=%5Cfrac%7B8%7D%7B%5Csqrt%7B2%7D%7D%3D%5Cfrac%7B%5Csqrt%7B2%7D%2Ax%20%7D%7B%5Csqrt%7B2%7D%7D%5C%5C%5C%5C%5Cfrac%7B8%7D%7B%5Csqrt%7B2%7D%7D%3Dx)
![\frac{\sqrt{2} }{\sqrt{2} } *\frac{8}{\sqrt{2}}\\\\\frac{8\sqrt{2}}{\sqrt{4}} \\\\\frac{8\sqrt{2}}{2}\\\\4\sqrt{2}=x](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B2%7D%20%7D%7B%5Csqrt%7B2%7D%20%7D%20%2A%5Cfrac%7B8%7D%7B%5Csqrt%7B2%7D%7D%5C%5C%5C%5C%5Cfrac%7B8%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B4%7D%7D%20%5C%5C%5C%5C%5Cfrac%7B8%5Csqrt%7B2%7D%7D%7B2%7D%5C%5C%5C%5C4%5Csqrt%7B2%7D%3Dx)
Simplify:
![x=4\sqrt{2}](https://tex.z-dn.net/?f=x%3D4%5Csqrt%7B2%7D)
6.) 10 is the hypotenuse. Insert values:
![10=\sqrt{2}*x](https://tex.z-dn.net/?f=10%3D%5Csqrt%7B2%7D%2Ax)
Divide
from both sides and rationalize:
![\frac{10}{\sqrt{2}}=\frac{\sqrt{2}*x}{\sqrt{2}} \\\\\frac{10}{\sqrt{2}} =x](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B%5Csqrt%7B2%7D%7D%3D%5Cfrac%7B%5Csqrt%7B2%7D%2Ax%7D%7B%5Csqrt%7B2%7D%7D%20%5C%5C%5C%5C%5Cfrac%7B10%7D%7B%5Csqrt%7B2%7D%7D%20%3Dx)
![\frac{\sqrt{2}}{\sqrt{2}}*\frac{10}{\sqrt{2}}\\\\\frac{10\sqrt{2}}{\sqrt{4}}\\\\\frac{10\sqrt{2}}{2}\\\\5\sqrt{2}=x](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B2%7D%7D%2A%5Cfrac%7B10%7D%7B%5Csqrt%7B2%7D%7D%5C%5C%5C%5C%5Cfrac%7B10%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B4%7D%7D%5C%5C%5C%5C%5Cfrac%7B10%5Csqrt%7B2%7D%7D%7B2%7D%5C%5C%5C%5C5%5Csqrt%7B2%7D%3Dx)
Simplify:
![x=5\sqrt{2}](https://tex.z-dn.net/?f=x%3D5%5Csqrt%7B2%7D)
7.) Draw the figure like the squares in problems 3 and 10. The problem says that the perimeter is 48, so divide 48 by 4, which is 12. A side is 12 meters (or a leg). The diagonal is the hypotenuse of a triangle. Insert values:
![x=\sqrt{2}*12](https://tex.z-dn.net/?f=x%3D%5Csqrt%7B2%7D%2A12)
Simplify:
![x=12\sqrt{2}](https://tex.z-dn.net/?f=x%3D12%5Csqrt%7B2%7D)
Finito.
To solve this, we have to complete the square of ![x^{2} +2x](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%2B2x)
To do this with halve the
and get rid of the x, then get rid of the power on
, and then put them all in brackets, and the square the bracket, like so:
becomes ![(x+1)^{2}](https://tex.z-dn.net/?f=%28x%2B1%29%5E%7B2%7D)
However
does not equal
If we expand
we get
instead.
So to make it equal, all we do is subtract 1.
So when we complete the square of
, we get ![(x+1)^{2}-1](https://tex.z-dn.net/?f=%28x%2B1%29%5E%7B2%7D-1)
---------------------------------------------------------
So
becomes:
![(x+1)^{2}-1-5=0](https://tex.z-dn.net/?f=%28x%2B1%29%5E%7B2%7D-1-5%3D0)
![(x+1)^{2}-6=0](https://tex.z-dn.net/?f=%28x%2B1%29%5E%7B2%7D-6%3D0)
![(x+1)^{2}=6](https://tex.z-dn.net/?f=%28x%2B1%29%5E%7B2%7D%3D6)
x + 1 = ±√6
x = -1 ± √6
______________________________
Answer:
x = -1 ± √6
Answer:
One 25 fl oz bottle costs $8
Step-by-step explanation:
144÷18=8