Question

Cho biểu thức : \(P=\left[2-\dfrac{2\sqrt{x}}{\sqrt{x}-3}+\dfrac{5\left(\sqrt{x}+4\right)}{x-9}\right]:\left(1-\dfrac{5}{\sqrt{x}+3}\right)\)

a, Rút gọn P

b, Tìm x để \(P< -\dfrac{1}{2}\)

c, Tìm MaxQ = \(P.\left(x\sqrt{x}-8x+15\sqrt{x}\right)\)

Answer 1

a: \(P=\dfrac{2x-18-2x-6\sqrt{x}+5\sqrt{x}+20}{x-9}:\dfrac{\sqrt{x}+3-5}{\sqrt{x}+3}\)

\(=\dfrac{-\sqrt{x}+2}{x-9}\cdot\dfrac{\sqrt{x}+3}{\sqrt{x}-2}=\dfrac{-1}{\sqrt{x}-3}\)

b: Để P<-1/2 thì P+1/2<0

=>\(\dfrac{-1}{\sqrt{x}-3}+\dfrac{1}{2}< 0\)

=>\(\dfrac{-2+\sqrt{x}-3}{2\left(\sqrt{x}-3\right)}< 0\)

=>\(\dfrac{\sqrt{x}-5}{2\left(\sqrt{x}-3\right)}< 0\)

=>3<căn x<5

=>9<x<25

c: \(Q=\dfrac{-1}{\sqrt{x}-3}\cdot\sqrt{x}\left(\sqrt{x}-3\right)\left(\sqrt{x}-5\right)=-x+5\sqrt{x}\)

\(=-\left(x-5\sqrt{x}+\dfrac{25}{4}-\dfrac{25}{4}\right)=-\left(\sqrt{x}-\dfrac{5}{2}\right)^2+\dfrac{25}{4}< =\dfrac{25}{4}\)

Dấu = xảy ra khi x=25/4