a.

ĐKXĐ: \(x\ge1\)

\(\sqrt{x-1}+\sqrt{x^3+x^2+x+1}=1+\sqrt{\left(x-1\right)\left(x^3+x^2+x+1\right)}\)

\(\Leftrightarrow\sqrt{x-1}\left(\sqrt{x^3+x^2+x+1}-1\right)-\left(\sqrt{x^3+x^2+x+1}-1\right)=0\)

\(\Leftrightarrow\left(\sqrt{x-1}-1\right)\left(\sqrt{x^3+x^2+x+1}-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x-1}=1\\\sqrt{x^3+x^2+x+1}=1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x^3+x^2+x=0\end{matrix}\right.\)

\(\Leftrightarrow...\)

b.

ĐKXĐ: \(x\ge-1\)

\(x^2-6x+9+x+1-4\sqrt{x+1}+4=0\)

\(\Leftrightarrow\left(x-3\right)^2+\left(\sqrt{x+1}-2\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-3=0\\\sqrt{x+1}-2=0\end{matrix}\right.\)

\(\Leftrightarrow x=3\)

c.

ĐKXĐ: \(-2\le x\le\dfrac{4}{5}\)

\(VT=2x+3\sqrt{4-5x}+1.\sqrt{x+2}\)

\(VT\le2x+\dfrac{1}{2}\left(9+4-5x\right)+\dfrac{1}{2}\left(1+x+2\right)=8\)

Dấu "=" xảy ra khi và chỉ khi \(x=-1\)

d.

ĐKXĐ: \(x>1\)

\(\Leftrightarrow\dfrac{x^2+x+1-1}{\sqrt{x^2+x+1}}=\dfrac{1-\left(x-1\right)}{\sqrt{x-1}}\)

Đặt \(\left\{{}\begin{matrix}\sqrt{x^2+x+1}=a>0\\\sqrt{x-1}=b>0\end{matrix}\right.\)

\(\Rightarrow\dfrac{a^2-1}{a}=\dfrac{1-b^2}{b}\)

\(\Leftrightarrow a-\dfrac{1}{a}=\dfrac{1}{b}-b\)

\(\Leftrightarrow a+b-\dfrac{a+b}{ab}=0\)

\(\Leftrightarrow\left(a+b\right)\left(1-\dfrac{1}{ab}\right)=0\)

\(\Leftrightarrow1-\dfrac{1}{ab}=0\)

\(\Leftrightarrow ab=1\)

\(\Leftrightarrow\left(x-1\right)\left(x^2+x+1\right)=1\)

\(\Leftrightarrow x^3-1=1\)

\(\Leftrightarrow x=\sqrt[3]{2}\)

Trả lời

x2−5x+14≐(x−3)2+x+5≥x+5≥x+1+4≥4x+1" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvea,arial,sans-serif; font-size:14px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">Ta có: \(x^2-5x+14=\left(x-3\right)^2+x+5\ge x+5\ge x+1+4\ge4\sqrt{x+1}\)x2−5x+14≐(x−3)2+x+5≥x+5≥x+1+4≥4x+1" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvea,arial,sans-serif; font-size:14px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">\(\Rightarrow VT\ge VP\)x2−5x+14≐(x−3)2+x+5≥x+5≥x+1+4≥4x+1" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvea,arial,sans-serif; font-size:14px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">Vậy để \(VT\ge VP\Leftrightarrow x=3\)(dấu "=" xảy ra)   

ĐKXĐ:x\(\ge-1\)

Đặt \(\sqrt{x+1}=a\ge0\)

\(\Rightarrow\hept{\begin{cases}a^2=x+1\\a^2-1=x\\x^2=a^4-2a^2+1\end{cases}}\)

Khi đó pt trên trở thành : \(4a=a^4-2a^2+1-5\left(a^2-1\right)+14\)

\(\Leftrightarrow a^4-2a^2+1-5a^2+5+14-4a=0\)

\(\Leftrightarrow a^4-7a^2-4a+20=0\)

\(\Leftrightarrow a^4-4a^2-3a^2+6a-10a+20=0\)

\(\Leftrightarrow a^2\left(a-2\right)\left(a+2\right)-3a\left(a-2\right)-10\left(a-2\right)=0\)

\(\Leftrightarrow\left(a-2\right)\left(a^3+2a^2-3a-10\right)=0\)

\(\Leftrightarrow\left(a-2\right)\left(a^3-2a^2+4a^2-8a+5a-10\right)=0\)

\(\Leftrightarrow\left(a-2\right)\left(a-2\right)\left(a^2+4a+5\right)=0\)

\(\Leftrightarrow\left(a-2\right)^2=0\)(vì a²+4a+5=(a+2)²+1\(\ge1>0\))

\(\Leftrightarrow x=2\)(thỏa mãn ĐKXĐ)

2.

\(DK:\hept{\begin{cases}x\ge-\frac{1}{5}\\x\ne0\end{cases}}\)

PT

\(\Leftrightarrow6+3\sqrt{5x+1}\left(\sqrt{5x+1}-1\right)=14\left(\sqrt{5x+1}-1\right)\)

\(\Leftrightarrow15x+23-17\sqrt{5x+1}=0\)

\(\Leftrightarrow\left(68-17\sqrt{5x+1}\right)+\left(15x-45\right)=0\)

\(\Leftrightarrow\frac{17\left(x-3\right)}{4+\sqrt{5x+1}}+15\left(x-3\right)=0\)

\(\Leftrightarrow\left(x-3\right)\left(\frac{17}{4+\sqrt{5x+1}}+15\right)=0\)

Vi \(\frac{17}{4+\sqrt{5x+1}}+15>0\)

\(\Rightarrow x=3\left(n\right)\)

Vay nghiem cua PT la \(x=3\)

a/ ĐKXĐ: \(x\ge2\)

\(\Leftrightarrow\sqrt{x+1}=1+\sqrt{x-2}\)

\(\Leftrightarrow x+1=1+x-2+2\sqrt{x-2}\)

\(\Leftrightarrow\sqrt{x-2}=1\)

\(\Leftrightarrow x=3\)

b/ ĐKXĐ: \(x^2\ge2\)

Đặt \(\sqrt{x^2-2}=t\ge0\Rightarrow x^2=t^2+2\)

Pt trở thành: \(t^2+2-t=4\)

\(\Leftrightarrow t^2-t-2=0\Rightarrow\left[{}\begin{matrix}t=-1\left(l\right)\\t=2\end{matrix}\right.\)

\(\Rightarrow\sqrt{x^2-2}=2\Leftrightarrow x^2=6\Rightarrow x=\pm\sqrt{6}\)

c/

\(\Leftrightarrow\sqrt{3\left(x+1\right)^2+4}+\sqrt{5\left(x+1\right)^2+9}=5-\left(x+1\right)^2\)

Do \(\left(x+1\right)^2\ge0\) ;\(\forall x\)

\(\Rightarrow\left\{{}\begin{matrix}\sqrt{3\left(x+1\right)^2+4}\ge\sqrt{0+4}=2\\\sqrt{5\left(x+1\right)^2+9}\ge\sqrt{0+9}=3\end{matrix}\right.\)

\(\Rightarrow VT\ge5\)

\(VP=5-\left(x+1\right)^2\le5\)

\(\Rightarrow VT\ge VP\)

Dấu "=" xảy ra khi và chỉ khi: \(\left(x+1\right)^2=0\Leftrightarrow x=-1\)

Tham khảo:

Giải pt: \(\sqrt{x-2} \sqrt{4-x}=2x^2-5x-1\) - Hoc24

Lời giải:

1. ĐKXĐ: $x\geq \frac{-5+\sqrt{21}}{2}$

PT $\Leftrightarrow x^2+5x+1=x+1$

$\Leftrightarrow x^2+4x=0$

$\Leftrightarrow x(x+4)=0$

$\Rightarrow x=0$ hoặc $x=-4$

Kết hợp đkxđ suy ra $x=0$

2. ĐKXĐ: $x\leq 2$

PT $\Leftrightarrow x^2+2x+4=2-x$

$\Leftrightarrow x^2+3x+2=0$

$\Leftrightarrow (x+1)(x+2)=0$

$\Leftrightarrow x+1=0$ hoặc $x+2=0$

$\Leftrightarrow x=-1$ hoặc $x=-2$
3.

ĐKXĐ: $-2\leq x\leq 2$

PT $\Leftrightarrow \sqrt{2x+4}=\sqrt{2-x}$

$\Leftrightarrow 2x+4=2-x$

$\Leftrightarrow 3x=-2$

$\Leftrightarrow x=\frac{-2}{3}$ (tm)

 

Bạn tham khảo lời giải tại đây:

https://hoc24.vn/cau-hoi/giai-pt-sqrtx-2sqrt4-x2x2-5x-1.219493072549