Page 39 - Aeronautical Engineer Data Book

26 Aeronautical Engineer’s Data Book
2.8.2 Logarithms
x
If N = a then log a N = x and N = a log a N
N
log b
log a N = 3
log a
b
log(ab) = log a + log b
��
a
log 3 = log a – log b
b
n
log a = n log a
1
n
� = 3 log a
log �a
n
log a 1 = 0
log e N = 2.3026 log 10 N
2.8.3 Quadratic equations
2
If ax + bx + c = 0
–b ± � ��
2
– 4
b
ac
x = 33
2a
2
2
If b –4ac > 0 the equation ax + bx + c = 0 yields
two real and different roots.
2
2
If b –4ac = 0 the equation ax + bx + c = 0 yields
coincident roots.
2
2
If b –4ac < 0 the equation ax + bx + c = 0 has
complex roots.
2
If and are the roots of the equation ax +
bx + c = 0 then
b
sum of the roots = + = – 3
a
c
product of the roots = = 3
d
2
The equation whose roots are and is x – (
+ )x + = 0.
2
Any quadratic function ax + bx + c can be
2
expressed in the form p (x + q) + r or r – p (x
2
+ q) , where r, p and q are all constants.
2
The function ax + bx + c will have a maximum
value if a is negative and a minimum value if a
is positive.