Page 40 - Aeronautical Engineer Data Book

Fundamental dimensions and units 27
2
2
If ax + bx + c = p(x + q) + r = 0 the minimum
value of the function occurs when (x + q) = 0
and its value is r.
2
2
If ax + bx + c = r – p(x + q) the maximum value
of the function occurs when (x + q) = 0 and its
value is r.
2.8.4 Cubic equations
3
2
x + px + qx + r = 0
3
x = y – 1 3 3 p gives y + 3ay + 2b = 0
3
where
2
1
3a = –q – 1 3 3 p 2 ,2b = p 3 –
33 pq + r
3
3
3 2 7 3
On setting
2
S = [–b + (b + a
3 1/2 1/3
) ]
and
2
) ]
T = [–b – (b + a 3 1/2 1/3
the three roots are
1
= S + T – 3 p
3
x 1
3
�
1
x = – 1 2 3 3 (S + T) + �3\2 i(S – T) – 3 p
3
2
3
�
1
x = – 1 2 3 3 (S + T) – �3\2 i(S – T) – 3 p.
3
3
3
For real coefficients
2
3
all roots are real if b + a ≤ 0,
2
3
one root is real if b + a > 0.
3
2
At least two roots are equal if b + a = 0.
Three roots are equal if a = 0 and b = 0. For b 2
3
+ a < 0
there are alternative expressions:
1
1
1
= 2c cos 1 3 3 – 3 px = 2c cos ( + 2π) –
3 p
3 3
3
3
x 1 2
3 3 3 3
1
1
= 2c cos 33 ( + 4π) – p
3 3
x 3
3 3
b
where c = –a and cos = – 33
2
c 3
2.8.5 Complex numbers
If x and y are real numbers and i = �–1� then
the complex number z = x + iy consists of the
real part x and the imaginary part iy.
z = x – iy is the conjugate of the complex
�
number z = x + iy.