CPhill

 3x^2 - 24x + 72      factor out  the 3

3 [ x^2  -  8x  +  24 ]

Take 1/2 of 8  = 4........square it =  16....add and subtract it

3 [ x^2 - 8x + 16 +  24 - 16 ]

Factor the first three terms and simplify

3 [ ( x - 4)^2 + 8 ]

3 (x - 4)^2  + 24

a + b + c  =

3 + (-4) + 24  =

-1 + 24

23

-8–4r < -4     add 8 to both sides

-4r <  4       divide both sides by -4....reverse the inequality sign

r >  -1

1–3u >  10         subtract 1 from both sides

-3u >  9           

Divide both sides by -3   and, since we're dividing by a negative, reverse the inequality sign

u < -3

70%  =  .70

So

.70 * 200  =

140

This would be true if we were setting the equation to 0  and solving

But we aren't solving anything here......we are just completing the square

Using

sin^2 ( theta) + cos^2 (theta)  = 1    we have

sin^2(theta) +  ( -12 / 13)^2  = 1

sin^(theta)  =  1 - (-12/13)^2

sin^2(theta)  =  1 - 144/169

sin^2(theta)  =    169/169  -  144/169

sin^2 (theta)  =  25/169       take the square root of both sides

sin (theta)  = ± √ [25/169]     and the sine is negative in the 3rd quadrant

sin (theta)  =  -5/13

tan (theta)  =   sin (theta) / cos (theta)  =  ( -5/13) / (-12/13)  =  5/12

x^2-3x+9 = x+41         subtract the right side from both sides

x^2 - 4x - 32  = 0        factor

(x - 8) (x + 4)  = 0

Setting each factor to 0  and solving for  x, we have that

x= 8   and x = -4

And the positive difference between these is

8 - (-4)  =

12

4s^2 + 28s + 45    factor out  the 4

4 [ s^2  + 7s  +  45/4]

Take (1/2) of 7  = 7/2......square this = 49/4  ....add it and subtract it

4 [ s^2 + 7s + 49/4 + 45/4 - 49/4 ] 

Factor the first three terms and simplify

4 [  ( s + 7/2)^2  - 1 ]

4(s + 7/2)^2  - 4

So q =  -4

3x^2 + x - 4      factor out the 3

3 [x^2 +  (1/3)x - (4/3) ]

Take (1/2) of (1/3)  = 1/6.......square this = 1/36......add it and subtract it

3 [ x^2 + (1/3)x  + 1/36 - (4/3) - 1/36 ]   factor the first 3 terms

3 [ ( x + 1/6)^2 - 1/36 - 48/36 ]

3 [ (x + 1/6)^2 - 49/36 ]

3(x + 1/6)^2 - 49/12

So k =  -49/12

Here's a ( highly questionable ) proceedure.......

1/sec * tan  =  sin

cos * sin/cos  = sin

sin  = sin                multiply both sides by the sin

sin^2  = sin^2

sin^2  =  1 - cos^2       add cos^2  to both sides

sin^2 + cos^2  =  1       divide through by cos^2

sin^2 / cos^2    + 1  =  1 / cos^1

tan^2  +  1  =  sec^2

Note :  multiplying/dividing on both sides of an identity isn't usually allowed......!!!!