Đáp án: `S={-3/4 ; 1/4 ; (3 \pm 2\sqrt5)/4}`
Giải thích các bước giải:
ĐK: `x+3/4 >= 0 <=> x >=-3/4`
`x+1+\sqrt(x+3/4) >=0 `
PT `<=> \sqrt(x+1+\sqrt(x+3/4)) = -x-1/4`
`<=> x+1+\sqrt(x+3/4) = (x+1/4)^2 (ĐK: -x-1/4 ≥ 0 <=> x ≤-1/4)`
`<=> x+1+\sqrt(x+3/4) =x^2+1/2 x+1/16`
`<=> \sqrt(x+3/4)=x^2-1/2x-15/16`
`<=> \sqrt(x+3/4) = (x-5/4)(x+3/4)`
`<=> \sqrt(x+3/4) - (x-5/4)(x+3/4)=0`
`<=> \sqrt(x+3/4) [1- \sqrt(x+3/4) (x-5/4)]=0`
TH1: `\sqrt(x+3/4)=0 <=> x=-3/4`
TH2: `1-\sqrt(x+3/4) (x-5/4)=0`
`<=> \sqrt(x+3/4) (x-5/4)=1`
`<=> (x+3/4)(x-5/4)^2 = 1`
`<=>` \(\left[ \begin{array}{l}x=\dfrac{1}{4}\\x=\dfrac{3 \pm 2\sqrt5}{4}\end{array} \right.\)
Vậy `S={-3/4 ; 1/4 ; (3 \pm 2\sqrt5)/4}`
