Maths - tangent

Let y = mx + c be the equation of the straight line.For it is tangent to the curve y = 2x² - 4x + 5 :
mx + c = 2x² - 4x + 5
2x² - (m + 4)x + 5 - c = 0 have double roots,
Δ = (m + 4)² - 8(5 - c) = 0
c = 5 - (m+4)²/8For it is tangent to the circle x² + y² - 12x + 32 = 0 :
x² + (mx + c)² - 12x + 32 = 0
x² + m²x² + 2mcx + c² - 12x + 32 = 0
(m² + 1)x² + (2mc - 12)x + c² + 32 = 0 have double roots,
Δ = 4(mc - 6)² - 4(m² + 1)(c² + 32) = 0
m²c² - 12mc + 36 = m²c² + 32m² + c² + 32
32m² + 12mc + c² - 4 = 0
32m² + 12m(5 - (m+4)²/8) + (5 - (m+4)²/8)² - 4 = 0
32m² + 60m - 12m(m+4)²/8 + (m+4)⁴/64 - 10(m+4)²/8 + 25 - 4 = 0
2048m² + 3840m - 96m(m+4)² + (m+4)⁴- 80(m+4)² + 1344 = 0
m⁴- 80m³ + 1296m² + 1920m + 320 = 0
m₁= - 1.185155153 , c₁= 4.009581061
m₂= - 0.191790155 , c₂= 3.187192222
m₃= 24.94911101 , c₃= - 99.75637853
m₄= 56.4278343 , c₄= - 451.4403948
∴ The equations of the straight line are
y = - 1.185155153 x + 4.009581061 ,
y = - 0.191790155 x + 3.187192222 ,
y = 24.94911101 x - 99.75637853 ,
y = 56.4278343 x - 451.4403948