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Remember, x = x(0) * 2^(-t/h),,,,where x(0) is initial amount at t=0,,t=time(yrs), and h = half life (yrs)
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x = x(0) *2^(-t/5.3),,,,,,,(a)
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subst in given values
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10 = 15 *2^( - t/5.3)
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10/15 =2/3 = 2^(-t/5.3)
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take log both sides,,,,,remember log (a^b) = b(loga)
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log (2/3) = log (2^(-t/5.3) ) = (-t/5.3)log2
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log(2/3) / log2 = (-t/5.3)
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-.585 = - t/5.3
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3.10 = t,,,,,,,,,,(b)
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check
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10 = 15 *2^(-3.1/5.3) =10,,,,,,ok
.

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-3-{ x-3x-y} -y
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-3 -1{x-3x-y} -y
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distribute (-1)
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-3 -x +3x +y -y
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-3 +2x
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2x-3
.

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9x^2 +4y^2
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very interesting problem.
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If this was ,, 9x^2 -4y^2 ,, just factor difference of squares,,,
(3x+2y)(3x-2y). To use this form on the positive, lets rewrite the eqn
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(9x^2 - (-4) y^2
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(3x+sqrt(-4)y) (3x- sqrt(-4)y)
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(3x+2iy) (3x - 2iy)
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check using FOIL
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F = 9x^2,,,,O= -6ixy,,,I = +6ixy ,,,,L = -4i^2 y^2 = 9x^2 +4y^2,,(i^2 = -1),,,,,ok

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20am -m +ma
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m(20a -1 +a)
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m( 21a -1)
.
21am -m
.

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x/(x+1) + 5/x = 1/ ( x^2 +x)
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LCD = x (x+1) = x^2 +x,,,,,,,,,,multiply thru
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x(x) + 5(x+1) = 1
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x^2 +5x +5 =1
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x^2 +5x +4 =0
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(x+1) (x+4) =0
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x = -1,,-4
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check
for (-4),,,,(-4)/(-4+1) +5/(-4) = 1/ { (-4)^2 +(-4) }
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(-4/ -3 ) - ( 5/4) = 1 / 12
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( 16 -15 ) /12 = 1/12
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1/12 = 1/12 ,,,,,,,,,,ok
..
for (-1),,,,,start with,,,,,x^2 +5x +4 =0,, (-1)^2 +5(-1) +4 =0,,,, 1 -5 +4 =0,,,,,,ok
.

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a^-1b^-1 / a^2 -b^2 / ab
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please check to assure correct problem.
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(a^-1 *b^-1) / (a^2 -b^2) / (ab)
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as with fractions, invert and multiply instead of divide
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( a^-1 * b^-1) * ( ab) / (a^2 -b^2)
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a^-1 = 1/a,,,,,,b^-1 = 1/b
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( 1/ab) * (ab) / ( a^2 -b^2)
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1/ ( a^2 - b^2 )

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Let x=short length,,,,y= long length
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(1) x + y = 20,,,,,y = 20 -x
(2) 4x = 3y-4
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substitute (1) in (2)
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4x = 3( 20-x) -4
4x = 60 -3x -4
7x =56
x=8
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substitute in (1)
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y= 20 -8
y=12
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check (1) 8+12 =20,,,,ok
(2) 4(8) = 3( 12) -4
32 =32,,,,,,,,,,,,,,,,,,,,,,ok

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Solving ALGEBRA problems,,, use order of operations in reverse,,,PEMDAS opposite
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[ 6-2/3y] -4 =8,,,,,,,,add 4 both sides
[ 6-2/3y] = 12,,,,,,,,,,subtract 6 both sides
-2/3y = 6,,,,,,,,,,,,,,,,,divide by - 2 and mult by 3,,both sides
y = - 6 (3/2),,,,,,,,,,,,,simplify
y= -9
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check
[ 6-2/3(-9)] -4=8
[6+6] -4 = 8
12-4=8
8=8,,,,,ok

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3^x -4 =2
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3^x = 6
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take log both sides
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log(3^x) =log 6
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or ,,,,,,x log 3 = log 6
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x= log 6 / log 3
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x= 1.631
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check,,,,,,3^(1.631) =6,,,,,ok
.
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if problem was ,,,,,3^(x-4) = 2
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(x-4) log 3 = log 2
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x-4 = log2/log3 = .631
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x= 4.631
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check
3^(4.631-4) =2,,,,, ok

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9y^2 -12y +4,,,,,,,,,factors of 9(9-1, 3-3) factors of 4(4-1, 2-2),,, (+) third term means signs are the same, and factor multiples add,,,,(-) 2nd term means, signs are (-).
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( 3y-2) (3y-2)
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F = 3y*3y= 9y^2
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O= 3y*-2 = -6y
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I = 3y *-2 = -6y
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L = -2*-2 = +4
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combining middle terms ,,, 9y^2 -12y +4
.

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7k^2 -9k -10,,,,,,,,,Factors of 7(7*1),,,factors of 10 (10*1,,,5*2),,,,,(-) last term means difference of factor multiples,,,,,,,,(-) middle term means largest factor multiple (-)
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(7k +5)(k-2)
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F = 7k*k = 7k^2
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O = 7K *-2 = -14 k
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I = k*5 = 5k
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L= 5*-2= -10
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combining middle terms,,,, 7k^2 -9k -10
.

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4n^2 -5n -6,,,,,factors of 4( 4*1, 2*2),, factors of 6 (6*1,,3*2),,,last term (-) means difference of
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middle terms, or factors,,,, 2nd term (-) means largest factor multiple is (-)
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(4n+3)(n-2)
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F = 4n *n = 4n^2
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O = 4n * -2 =-8n
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I = n*3 =3n
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L = 3*-2 = -6
.
combining middle terms ,,,,4n^2 -5n -6

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10 (y-6)
.
10*y -10*6
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10y -60
.

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please see problem # 200851,,,

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(x+6) / x = -5
cross multiply
x+6 = -5x
6x +6 =0
6(x+1) =0
x=-1
check
(-1) +6 = -5 (-1)
5=5,,,,,,,ok

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(1) 4a +3b =27
(2) 3a +4b = 29
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multiply (1) by 3,,,,and (2) by 4
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(3) 12a +9b = 81
(4) 12a +16b = 116
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(4) - (3)
7b = 35
b=5
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subst into (1)
4a +3(5) = 27
4a =12
a=3
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check
(1) 4(3) +3(5) = 27,,,,,12 +15 =27 ,,,,,,ok
(2) 3(3) +4(5) =29,,,,9+20=29,,,,,,,ok
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subst into 7a + 7b/2 = 7(3) +7(5) /2 = 21 + 35/2 = 38.5
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or subst (7a +7b )/2 = (7*3 +7*5)/2= 56 /2 =28

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25 y^2 -30y +9
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(5y-3)(5y-3)
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F = (5y)(5y) = 25 y^2
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O = (5y)(-3) = -15y
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I = ((5y)(-3) = -15y
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L = (-3)(-3) = +9
.
combining middle terms ,,,,,, 25y^2 -30y + 9,,,,,ok

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Very good question.
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Decimal to fraction
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.5 = 5/10 = 1/2
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.25 = 25 /100 = 1/4
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.375 = 375 / 1000 = 3/8 ,,,,,,,( divided by 125)
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To generalize, set decimal number over place value
.
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Fraction to decimal
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Remember that 1/2 means "1 divided by 2" = .5
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1/4 = 1 divided by 4 = .25
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3/8 = 3 divided by 8 = .375
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also mixed numbers, either change to improper fraction and proceed as above, or just do fraction part and add back in whole number part.

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(3sqrt3)^4 ,,,,,,,,check = 729
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[ 3 * 3^1/2] ^4
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[ 3^1 1/2]^4
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[3^ 3/2]^4
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3^ (12/2)
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3^6= 729,,,,,ok

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[ { x^2-6x +9} / {12-4x} ] * [ { x^6 -9x^4} / {x^3 -3x^2}]
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[ ( x-3)(x-3) / -4x (x-3) ] * [ ( x^4)(x^2 -9) / (x^2) ( x-3) ]
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simplify by cancelling like terms, num and den
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-x^4 (x^2 -9) / 4 x^2
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(-1/4) x^2 ( x+3)(x-3)
.
Check using x=2 in both original and answer, both = +5, ok

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a^2 -40 =0
.
a^2 =40
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sqrt both sides
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a= =/- sqrt 40 = +/- 6.325
.
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w^2 +49 =0
.
w^2 =-49
.
w = +/- sqrt -49 = +/- 7 i
.

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-3x^2 +7x +0 =0
,
a= (-3),,,,,,b= (7),,,,,,,,,c=0
.
Discriminant = b^2 -4ac = (7)^2 - 4 (-3) (0) = 49
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D > 0 ,,,,,therefore 2 real numbers as solutions,,,,,,0,,,2.3333
.
if D = 0,,,,,, one solution
.
if D less than 0 = no real solutions

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x^2 +x +sqrtx^2 +x -2 = 0
.
x^2 +x+x+x-2=0
.
x^2 +3x -2=0
.
use Quadratic Formula
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a=1,,,,b= 3,,,,c=-2
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x=[ -(3) +/- sqrt { (3)^2 -4 (1) (-2) } ] / 2(1)
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x= [ -3 +/- sqrt { 9 +8 } ] /2
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x= [-3 +/- 4.123 ] /2
.
x= .5615,,, -3.562
.
check by subst into original, with calc,,,,ok

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b^4 +13b^2 +36 =0
let a = b^2
a^2 +13a +36 =0
(a+9)(a+4) =0
subst back
(b^2 +9)(b^2 +4) =0
.
b^2 +9 =0,,,,, b^2 = -9,,,b= +/- 3i
.
b^2 +4 = 0,,,,,,b^2 = -4,,,,,,,,,b= +/- 2i

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w^2 +49
(w+7i)(w-7i)

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x^3 -4x
x(x^2 -4)
x(x+2)(x-2)

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The Quadratic Formula is the answer to
.
ax^2 +bx + c =0,,,,,, most books have a derivation
.
as to why the Discriminant (D) works, let's look at quadratic answer
.
x= [ - (b) +/- sqrt { (b^2) - 4 (a) (c) } ] / 2a
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or,,,,x= [-(b) +/- sqrt D ] / 2a
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there are 3 conditions for the discriminant (D)
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(1) D =0,,,,then x=[ -(b) +/- sqrt(0) ] /2a = -b/2a,,,,,therefore one answer
.
(2) D>0,,,,,,then x= [ -(b) +/- sqrt D ] /2a or x= [-(b) + sqrtD]/2a,,and x = [-b -sqrtD] /2a,,, or two answers
.
(3) D less than 0,,,,,,then x=[ -(b) +/- sqrt D ] /2a = [-b +/-sqrt (-#) ] /2a ,,,and with sqrt of a negative number, the answer is a complex number, not a real number,,,,,,therefore NO real number answers
.

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[ { X^2 -x -20} / {x^2-25} ] / [ {x^2 -16} / { x^2 -x -30} ]
.
As with fractions, to divide invert and multiply,,,a/b/c/d = a/b * d/c
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[ { x^2 -x-20} / {x^2 -25}] * [ { x^2 -x -30} / {x^2 -16} ]
.
factor and simplify by cancelling like terms
.
[ { (x-5)(x+4)} / { (x+5)(x-5) } ] * [ { (x-6)(x+5) } / { ( x+4)(x-4)} ]
.
( x-6) / (x-4),,,,,answer
.
check,, let x=1,,,,in original = 1.5,,,,,,,in answer = 1.5,,,,,,ok

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Remember order of operations, PEMDAS, Parenthesis, Exponents, Multiply / Divide, lt to rt, and ADD / Subtract, lt to rt
.
2+4 -6 -3*2 +3*7
.
2+4 -6 -6 +21
.
6 -6 -6 +21
.
0-6 +21
.
-6 +21
.
15

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(1) x+y= 152
(2) x/y = 5/2 = 2.5,,,,,,,x=(2.5y)
.
subst (2) into (1)
.
2.5y + y = 152
3.5 y =152
y= 43.43,,,,,,answer
.
subst in (2)
x = 2.5 * 43.43 = 108.57
.
check
(1) 108.57 +43.43 = 152,,,,ok
(2) 108.57 / 43.43 = 2.5,,,,,,ok

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(1) x +y =11
(2) x^2 + y^2 =73
.
from(1) y = (11-x)
.
subst into (2)
.
x^2 +(11-x)^2 =73
x^2 + (121 -22x +x^2 = 73
2x^2 -22x +48 =0
divide by 2
x^2 -11x +24 =0
factor
(x-8)(x-3) =0
x= 8 , 3
.
check
(1) 8+3=11,,,,ok
(2) 64 + 9 = 73 ,,,,,ok