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Answers by Michael F.

(5x^2)-(2x-3)^2 (answer)

5x2-(2x-3)2=5x2-(4x2-12x+9)=5x2-4x2+12x-9=x2+12x-9 That's as far as I can go without seeing the choices

y=x2+6x-4=(x+3)2-13 y+13=(x+3)2 vertex at (-3,-13) , y-(-13-n)=(x-(-3-m))2 is this parabola shifted m to right, n up.  New vertex at (5,-6) means that -13-n=-6 so n=-7, -3-m=5 so m=-8 and equation is y+6=(x-5)2 m/n=8/7  

It is necessary to assume that x1 and x0 are known.  This makes the fact that x2=x1-x0 given. We write xn+1=xn-xn-1 xn+2=xn+1-xn=xn-xn-1-xn=-xn-1 xn+3=xn+2-xn+1=-xn xn+4=xn+3-xn+2=-xn+xn-1 xn+5=xn+4-xn+3=xn-1 xn+6=xn+5-xn+4=xn xn+7=xn+6-xn+5=xn-xn-1=xn+1 ...

Let S be the sum in question. Each term is 5n-3 n=1  5×1-3=2 n=2  5×2-3=7, etc We write n terms both forward and backward as Note that the next to last term 5(n-1)-3=5n-8 S=2  +       7  +   12  + ....

Let r= # of bowls of rice b=# of bowls of broth f=# of bowls of fowl 2r=3b=4f=# of guests r+b+f=65 In terms of r we have b=(2/3)r, f=(1/2)r and so r+(2/3)r+(1/2)r=65 solving for r we get r=30 and so # of guests = 60  

a is a first quadrant angle. sin(a)=8/17, the numbers are two sides of an 8,15,17 right triangle. cos(a)=15/17 b is a second quadrant angle. tan(b)=-7/24, the numbers are two sides of a 7,24,25 right triangle.  The -7 is along the x-axis, the 24 is parallel to the y-axis. sin(b)=24/25,...

(sin2∅-1)/(tan2∅) = (1/sin2∅)-1 left side= -(1-sin2∅)/tan2∅ left side = -cos2∅/tan2∅ right side = (1/sin2∅)-1=(1-sin2∅)/sin2∅ right side = cos2∅/sin2∅=1/tan2∅ This is NOT an identity. Perhaps it needs revision.

3|m=11²-5k  Let m=3n 11²-3n=5k 121-3×2=115=5×23    m=6, k=23 121-3×7=100=5×20   m=21, k=20 121-3×12=85=5×17   m=36, k=17 121-3×17=70=5×14   m=51, k=14 121-3×22=55=5×11   m=66, k=11 121-3×27=40=5×8   ...

Find the area (answer)

This area is a triangle with the base along the y-axis from y=0 to y=4. The altitude is the distance of the intersection of the lines y=x and y=4-3x, the point (1,1), from the y-axis namely 1.  The area is 4×1/2=2. If you absolutely need to use calculus Area is ∫01(4-4x)dx=...

The pattern you seem to be looking at is for every integer n, the number g you seek is such that n/g = n+g This gives g2+ng-n=0 or that g=(-n±√(n2+4n))/2 If you use the plus sign you get n= 1     g=0.618033989 n=2     ...