\(\sqrt{5x-x^2}+2x^2-10x+6=0\)

ĐKXĐ : \(0\le x\le5\)

<=> \(\sqrt{5x-x^2}-2\left(5x-x^2\right)+6=0\)

Đặt \(\sqrt{5x-x^2}=t\)( t ≥ 0 ) ta được phương trình :\(t-2t^2+6=0\)(*)

Δ = b² - 4ac = 1 + 48 = 49

Δ > 0 nên (*) có hai nghiệm phân biệt t₁ = -3/2 (ktm) ; t₂ = 2 (tm)

=> \(\sqrt{5x-x^2}=2\)

<=> 5x - x² = 4 ( bình phương hai vế )

<=> x² - 5x + 4 = 0 (1)

Dễ thấy (1) có a + b + c = 1 - 5 + 4 = 0 nên có hai nghiệm phân biệt x₁ = 1 (tm) ; x₂ = c/a = 4 (tm)

Vậy phương trình đã cho có hai nghiệm x₁ = 1 ; x₂ = 4

a, Ta có : 

\(P=\frac{2x-3\sqrt{x}-2}{\sqrt{x}-2}=\frac{2x+\sqrt{x}-4\sqrt{x}-2}{\sqrt{x}-2}\)sử dụng tam thức bậc 2 khai triển biểu thức trên tử nhé 

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

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

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

b, Ta có : \(P=Q\)hay \(2\sqrt{x}+1=x-1\Leftrightarrow-x+2\sqrt{x}+2=0\)

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

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

TH1 : \(\sqrt{x}=1+\sqrt{3}\Leftrightarrow x=\left(1+\sqrt{3}\right)^2=1+2\sqrt{3}+3=4+2\sqrt{3}\)

TH2 : \(\sqrt{x}=1-\sqrt{3}\Leftrightarrow x=\left(1-\sqrt{3}\right)^2=1-2\sqrt{3}+3=4-2\sqrt{3}\)

Vậy \(x=4+2\sqrt{3};x=4-2\sqrt{3}\)thì P = Q 

んuﾘ ｲ giải pt vô tỉ không xét ĐK là tai hại :))

 \(P=\frac{2x-3\sqrt{x}-2}{\sqrt{x}-2}=\frac{2x-4\sqrt{x}+\sqrt{x}-2}{\sqrt{x}-2}\)

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

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

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

Để P = Q thì \(2\sqrt{x}+1=x-1\)( x ≥ 1 ; x ≠ 4 )

<=> \(x-2\sqrt{x}-2=0\)

<=> \(\left(\sqrt{x}-1\right)^2-3=0\)

<=> \(\left(\sqrt{x}-1-\sqrt{3}\right)\left(\sqrt{x}-1+\sqrt{3}\right)=0\)

<=> \(\orbr{\begin{cases}x=1+\sqrt{3}\\x=1-\sqrt{3}\end{cases}}\Rightarrow\orbr{\begin{cases}x=4+2\sqrt{3}\left(tm\right)\\x=4-2\sqrt{3}\left(ktm\right)\end{cases}}\)

Vậy với \(x=4+2\sqrt{3}\)thì P = Q

a)

+) P=2x−3√x−2√x−2 

=(2√x+1)(√x−2)√x−2 

=2√x+1.

+) Q=√x3−√x+2x−2√x+2 

=(√x+2)(x−1)√x+2 

=x−1.

b) Để $P=Q$ thì 2√x+1=x−1

Coi phương trình là phương trình bậc 2 của $\sqrt{x}$, chú ý chọn nghiệm dương của phương trình.

Đáp số: x=4+2√3 thì $P=Q$.

 Câu hỏi thuộc chủ đề: Rút gọn biểu thức chứa căn bậc hai

o l m . v n

ĐKXĐ : x ≥ 0

<=> \(x-5\sqrt{x}+2\sqrt{x}-10=0\)

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

<=> \(\left(\sqrt{x}-5\right)\left(\sqrt{x}+2\right)=0\)(1)

Vì \(\sqrt{x}+2\ge2>0\forall x\ge0\)

nên (1) <=> \(\sqrt{x}-5=0\)<=> \(\sqrt{x}=5\)<=> x = 25 (tm)

Vậy pt có nghiệm x = 25

ĐK: $x\ge 0$

x-3\sqrt{x}-10=0x−3x​−10=0

Đặt \sqrt{x}=t\left(t\ge0\right)x​=t(t≥0). Khi đó phương trình trở thành t^2-3t-10=0t2−3t−10=0

\Leftrightarrow\left(t^2-5t\right)+\left(2t-10\right)=0\Leftrightarrow\left(t+2\right)\left(t-5\right)=0⇔(t2−5t)+(2t−10)=0⇔(t+2)(t−5)=0

\Leftrightarrow\left[{}\begin{matrix}t+2=0\\t-5=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}t=-2\left(l\right)\\t=5\left(n\right)\end{matrix}\right.⇔[t+2=0t−5=0​⇔[t=−2(l)t=5(n)​

Với t = 5 ta có \sqrt{x}=5\Leftrightarrow x=25\left(tmđk\right)x​=5⇔x=25(tmđk)

Vậy phương trình có nghiệm x = 25.

ĐK: \(x\ge0\)

\(x-3\sqrt{x}-10=0\)

\(\Leftrightarrow x-5\sqrt{x}+2\sqrt{x}-10=0\)

\(\left(\sqrt{x}-5\right)\left(\sqrt{x}+2\right)=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x}-5=0\\\sqrt{x}+2=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=25\left(Tm\right)\\\sqrt{x}=-2\left(Loại\right)\end{matrix}\right.\)

Ta có \(\left(x+y\right)^2=xy+3y-1\)

<=>\(x^2+1=-y^2-xy+3y\)

Thế vào phương trình 2 ta có

\(x+y=1+\frac{y}{-y^2-xy+3y}\)

<=> \(x+y=1-\frac{1}{x+y-3}\)

Đặt x+y=a

=> \(a=1-\frac{1}{a-3}\)<=> \(a^2-4a+4=0\)=> a=2

=> x+y=2

Thế vào 1 ta có

\(4=y\left(2-y\right)+3y-1\)=> \(y^2-5y+5=0\)=> \(\orbr{\begin{cases}y=\frac{5+\sqrt{5}}{2}\\y=\frac{5-\sqrt{5}}{2}\end{cases}}\)

Vậy \(\left(x,y\right)=\left(-\frac{1+\sqrt{5}}{2},\frac{5+\sqrt{5}}{2}\right),\left(\frac{-1+\sqrt{5}}{2},\frac{5-\sqrt{5}}{2}\right)\)

a) Ta có : \(x=\sqrt[3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}+\sqrt[3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}\)

\(\Rightarrow x^3=2a+3.\sqrt[3]{a^2-\left(\frac{a+1}{3}\right)^2\left(\frac{8a-1}{3}\right)}.x\)

\(=2a+3\sqrt[3]{a^2-\frac{\left(a^2+2a+1\right)\left(8a-1\right)}{27}}.x\)

\(=2a+3\sqrt[3]{\frac{27a^2-\left(8a^3+15a^2+6a-1\right)}{27}}.x\)

\(=2a+3\sqrt[3]{\frac{-8a^3+12a^2-6a+1}{27}}.x\)

\(=2a+3x.\sqrt[3]{\frac{\left(1-2a\right)^3}{3^3}}=2a+3x.\frac{1-2a}{3}=2a+x\left(1-2a\right)\)

\(\Rightarrow x^2-2a+x\left(2a-1\right)=0\)\(\Leftrightarrow x^3-2a+2ax-x=0\Leftrightarrow x\left(x-1\right)\left(x+1\right)+2a\left(x-1\right)=0\)

\(\Leftrightarrow\left(x-1\right)\left(x^2+x+2a\right)=0\Leftrightarrow\orbr{\begin{cases}x-1=0\\x^2+x+2a=0\end{cases}}\)

Vì \(a>\frac{1}{8}\) nên \(x^2+x+2a>0\Rightarrow\)vô nghiệm.

Vậy x - 1 = 0  => x = 1 thoả mãn x là số nguyên dương.

b) \(\sqrt[3]{x+24}+\sqrt{12-x}=6\) (ĐKXĐ : \(x\le12\))

\(\Leftrightarrow\sqrt[3]{x+24}=6-\sqrt{12-x}\Leftrightarrow x+24=\left(6-\sqrt{12-x}\right)^3\)

\(\Leftrightarrow x+24=6^3-3.6^2.\sqrt{12-x}+3.6.\left(12-x\right)-\left(\sqrt{12-x}\right)^3\)

\(\Leftrightarrow x+24=216-108\sqrt{12-x}+216-18x-\sqrt{12-x}^3\)

\(\Leftrightarrow-19\left(12-x\right)+108\sqrt{12-x}+\sqrt{12-x}^3-180=0\)

 Đặt \(y=\sqrt{12-x},y\ge0\) . Phương trình trên tương đương với : 

\(-19y^2+108y+y^3-180=0\Leftrightarrow\left(y-10\right)\left(y-6\right)\left(y-3\right)=0\)

=> y = 10 (TM) hoặc y = 6 (TM) hoặc y = 3 (TM)

Vậy tập nghiệm của phương trình : \(S=\left\{-88;-24;3\right\}\)

a, Với \(x>0;x\ne4;x\ne9\)

\(A=\left(\frac{4\sqrt{x}}{2+\sqrt{x}}+\frac{8x}{4-x}\right):\left(\frac{\sqrt{x}-1}{x-2\sqrt{x}}-\frac{2}{\sqrt{x}}\right)\)

\(=\left(\frac{4\sqrt{x}\left(2-\sqrt{x}\right)+8x}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\right):\left(\frac{\sqrt{x}-1-2\left(\sqrt{x}-2\right)}{\sqrt{x}\left(\sqrt{x}-2\right)}\right)\)

\(=\left(\frac{8\sqrt{x}-4x+8x}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\right):\left(\frac{\sqrt{x}-1-2\sqrt{x}+4}{\sqrt{x}\left(\sqrt{x}-2\right)}\right)\)

\(=\frac{8\sqrt{x}+4x}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}:\frac{-\sqrt{x}+3}{\sqrt{x}\left(\sqrt{x}-2\right)}\)

\(=\frac{4\sqrt{x}\left(2+\sqrt{x}\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}:\frac{3-\sqrt{x}}{\sqrt{x}\left(2-\sqrt{x}\right)}=\frac{4\sqrt{x}}{2-\sqrt{x}}.\frac{\sqrt{x}\left(2-\sqrt{x}\right)}{3-\sqrt{x}}=\frac{4x}{3-\sqrt{x}}\)

b, Ta có : A = -2 hay 

\(\frac{4x}{3-\sqrt{x}}=-2\Rightarrow4x=-6+2\sqrt{x}\)

\(\Leftrightarrow4x+6-2\sqrt{x}=0\Leftrightarrow2\left(2x+3-\sqrt{x}\right)=0\)

\(\Leftrightarrow2x+3-\sqrt{x}=0\Leftrightarrow\sqrt{x}=2x+3\)

bình phương 2 vế ta có : 

\(x=\left(2x+3\right)^2=4x^2+12x+9\)

\(\Leftrightarrow-4x^2-11x-9=0\)giải delta ta thu được : \(x=-\frac{11\pm\sqrt{23}i}{8}\)

\(a,A=\left(\frac{4\sqrt{x}}{2+\sqrt{x}}+\frac{8x}{4-x}\right):\left(\frac{\sqrt{x}-1}{x-2\sqrt{x}}-\frac{2}{\sqrt{x}}\right)\)              

\(=\left(\frac{4\sqrt{x}}{2+\sqrt{x}}+\frac{8x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}\right):\left(\frac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-2\right)}-\frac{2}{\sqrt{x}}\right)\)

\(=\frac{4\sqrt{x}.\left(2-\sqrt{x}\right)+8x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}:\frac{\sqrt{x}-1-2.\left(\sqrt{x}-2\right)}{\sqrt{x}\left(\sqrt{x}-2\right)}\)

\(=\frac{8\sqrt{x}-4x+8x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}.\frac{\sqrt{x}\left(\sqrt{x}-2\right)}{\sqrt{x}-1-2\sqrt{x}+4}\)

\(=\frac{\left(4x+8\sqrt{x}\right)\left(\sqrt{x}\right)\left(\sqrt{x}-2\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)\left(-\sqrt{x}+3\right)}\)

\(=\frac{-4\sqrt{x}\left(\sqrt{x}+2\right)\left(\sqrt{x}\right)\left(2-\sqrt{x}\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)\left(-\sqrt{x}+3\right)}\)

\(=\frac{4x}{\sqrt{x}-3}\)

\(A=\left(\frac{4\sqrt{x}}{2+\sqrt{x}}+\frac{8x}{4-x}\right)\div\left(\frac{\sqrt{x}-1}{x-2\sqrt{x}}-\frac{2}{\sqrt{x}}\right)\)

\(=\left(\frac{4\sqrt{x}\left(2-\sqrt{x}\right)}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}+\frac{8x}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\right)\div\left(\frac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-2\right)}-\frac{2\left(\sqrt{x}-2\right)}{\sqrt{x}\left(\sqrt{x}-2\right)}\right)\)

\(=\frac{8\sqrt{x}-4x+8x}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\div\frac{-\sqrt{x}+3}{\sqrt{x}\left(\sqrt{x}-2\right)}\)

\(=\frac{4\sqrt{x}}{2-\sqrt{x}}\times\frac{\sqrt{x}\left(\sqrt{x}-2\right)}{-\sqrt{x}+3}=\frac{4\sqrt{x}}{2-\sqrt{x}}\times\frac{\sqrt{x}\left(2-\sqrt{x}\right)}{\sqrt{x}-3}=\frac{4\sqrt{x}}{\sqrt{x}-3}\)

Để A = -2 thì \(\frac{4\sqrt{x}}{\sqrt{x}-3}=-2\)

=> \(4\sqrt{x}=-2\sqrt{x}+6\)

<=> \(6\sqrt{x}=6\)<=> \(\sqrt{x}=1\)<=> x = 1 ( tm )

Vậy với x = 1 thì A = -2