中四數學題一問

Answer: 4/5

Solutions:

5 = 4a - 3a^2 = 4b - 3b^2
=> 3a^2 - 4a + 5 = 0 and 3b^2 - 4b + 5 = 0
thus a and b are the roots of the quadratic equation 3x^2 - 4x + 5 = 0

So, a + b = sum of roots = 4/3
and ab = product of roots = 5/3

therefore, 1/a + 1/b
= (a + b)/(ab)
= (4/3)/(5/3)
= 4/5

If a not equal to b，and given that 5=4a－3a^2=4b－3b^2，then 1/a + 1/b =?
Sol
5=4a－3a^2=4b－3b^2
3a^2－4a+5=0，3b^2－4b+5=0，a<>b
So
a，b為3x^2－4x+5=0之二根
1/a，1/b為3(1/x)^2-4(1/x)+5=0之二根
5x^2－4x+3=0
So
1/a+1/b=－(－4)/5=4/5