CPhill

\(\frac{x+4}{x+5} = \frac{x-3}{2} + \frac{x + 7}{5} \)

(x + 4) / ( x + 5)  =    [5(x-3) + 2(x + 7)] / 10         cross-multiply

10 ( x + 4)  =  (x + 5) ( 7x - 1)

10x + 40  =  7x^2 + 34x  - 5

7x^2 + 24x - 45  =  0

x =  [ -24 +/-  sqrt [ 24^2 + 45 * 4 * 7] ] / 14

x =  [-24 +/-  6 sqrt 51 ] 14  =   [ -12 +/-  3sqrt 51] / 7

By Heron's Formula

[ ABC ]  = sqrt [ 17 * 2 * 8 * 7]  =  4sqrt 119

Shortest altitude is always drawn to  the longest side

So

4sqrt 119  = (1/2) 15 * altitude

4sqrt 119  = 7.5 *altitude

altitude  =  (4 / 7.5)sqrt 119  ≈ 5.82

Angle B  =90

So ABC is right

The hypotenuse,  AC  = 12

So  AB  , CB  =   12/sqrt 2

AD = BD

[ ABD ]  = [ ABC ]  / 2  = (1/2) (1/2) (12/sqrt 2)^2  = (1/4) (144) / 2 =  18

Slope =  (a^2 - b^2)  / ( a -b)  =  [ (a + b) (a -b) ] / (a -b)  =  2 →  a + b =   2

Simplify as

4x^2 - 8x + 4y^2 + 14y  =  -12             complete the sqiare on  x, y

4(x^2 - 2x + 1) + 4(y^2 + [7/2] y  + 49/ 16)  =   -12 + (4)(1)  + 4 (49/16)

4 ( x-1)^2  + 4(y + 7/2)^2 =  17/4

The radius =  sqrt (17)  / 2

https://web2.0calc.com/questions/right-triangles_33

6x + 3y    has no min  value

Rearrange as

3a + 2b  = 2

9a + (2 + m)b  = k

We want the equations to match...multiply the first  equation by 3

9a + 6b = 6

Then

(2 + m)  = 6

m =4

And k =6

2(-1)^2 -8(-1) + 1 =  11

2(2)^2 - 8(2) + 1 =  -7

The function is continuous on [ -1, 2]  so the range is  [  -7 , 11 ]