a) ĐKXĐ: x khác 4, x >= 0
Ta có: x + √x - 6 = x + 3√x - 2√x - 6 = √x.(√x + 3) - 2.(√x + 3) = (√x - 2)(√x + 3)
P = đề
= [ -(√x - 3).(√x + 3) + (√x - 2)^2 - 9 + x ] / [ (√x - 2)(√x + 3) ]
= (-x + 9 + x - 4√x + 4 - 9 + x) / (√x - 2)(√x + 3)
= (x - 4√x + 4) / (√x - 2)(√x + 3)
= (√x - 2)^2 / (√x - 2)(√x + 3)
= (√x - 2) / (√x + 3)
b) P = 7/12
<=> (√x - 2)/(√x + 3) = 7/12
<=> 12√x - 24 = 7√x + 21
<=> 5√x = 45
<=> √x = 9
<=> x = 81
c) P > 1/2
<=> (√x - 2) / (√x + 3) > 1/2
<=> 2.(√x - 2) > √x + 3
<=> 2√x - 4 > √x + 3
<=> √x > 7
<=> x > 49
d) 1/P nguyên
<=> (√x + 3) chia hết (√x - 2)
<=> (√x - 2 + 5) chia hết cho (√x - 2)
<=> 5 chia hết cho √x - 2
<=> (√x - 2) thuộc Ư(5) = {0; 1; 5}
_ √x - 2 = 0 <=> x = 4 (loại)
_ √x - 2 = 1 <=> √x = 3 <=> x = 9
_ √x - 2 = 5 <=> √x = 7 <=> x = 49