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\(\:x\left(x^2+x+1\right)-x^2\left(x+1\right)-x+5\)
\(=x^3+x^2+x-x^3-x^2-x+5=5\)
Vậy biểu thức ko phụ thuộc vào biến x
\(\left(x^3+x^2y+xy^2+y^3\right)\left(x-y\right)+2x^4\)
\(=x^4-x^3y+x^3y-x^2y^2+x^2y^2-xy^3+y^3x-y^4+2x^4\)
\(=3x^4-y^4\)
a) 2x2 - 4x + 5
= 2( x2 - 2x + 1 ) + 3
= 2( x - 1 )2 + 3 ≥ 3 > 0 ∀ x ( đpcm )
b) 3x2 + 2x + 1
= 3( x2 + 2/3x + 1/9 ) + 2/3
= 3( x + 1/3 )2 + 2/3 ≥ 2/3 > 0 ∀ x ( đpcm )
c) -x2 + 6x - 10
= -x2 + 6x - 9 - 1
= -( x2 - 6x + 9 ) - 1
= -( x - 3 )2 - 1 ≤ -1 < 0 ∀ x ( đpcm )
d) -x2 + 3x - 3
= -x2 + 3x - 9/4 - 3/4
= -( x2 - 3x + 9/4 ) - 3/4
= -( x - 3/2 )2 - 3/4 ≤ -3/4 < 0 ∀ x ( đpcm )
e) \(\frac{x^2+4x+5}{2}>0\)
Vì 2 > 0
=> x2 + 4x + 5 > 0
=> x2 + 4x + 4 + 1 > 0
=> ( x + 2 )2 + 1 > 0 ( đúng )
=> \(\frac{x^2+4x+5}{2}>0\)∀ x ( đpcm )
f) \(\frac{-6+2x-x^2}{x^2+1}< 0\)
Vì x2 + 1 ≥ 1 ∀ x
=> -6 + 2x - x2 < 0
=> -x2 + 2x - 1 - 5
= -( x2 - 2x + 1 ) - 5
= -( x - 1 )2 - 5 < 0 ( đúng )
=> \(\frac{-6+2x-x^2}{x^2+1}< 0\)∀ x ( đpcm )
a,Ta có :\(2x^2-4x+5=\left(x^2-2x+1\right)+\left(x^2-2x+1\right)+3\)
\(=\left(x-1\right)^2+\left(x-1\right)^2+3=2\left(x-1\right)^2+3\)
Do \(2\left(x-1\right)^2\ge0\Leftrightarrow2\left(x-1\right)^2+3\ge3\forall x\inℝ\)
Hay :\(2x^2-4x+5>0\)
Vậy nên BPT luôn đúng với mọi số thực x
b,Ta có : \(3x^2+2x+1=x^2+2x+1+2x^2\)
\(=\left(x+1\right)^2+2x^2\)
Do \(\hept{\begin{cases}\left(x+1\right)^2\ge0\forall x\inℝ\\2x^2\ge0\forall x\inℝ\end{cases}}\Leftrightarrow\left(x+1\right)^2+2x^2\ge0\forall x\inℝ\)
Vậy nên BPT luôn đúng với mọi số thực x
c,Ta có : \(-x^2+6x-10=-\left(x^2-6x+10\right)\)
\(=-\left(x^2-6x+9\right)-1=-\left(x-3\right)^2-1\)
Do \(\left(x-3\right)^2\ge0\forall x\inℝ\Leftrightarrow-\left(x-3\right)^2-1\le-1\forall x\inℝ\)
Hay \(-x^2+6x-10\le-1\forall x\inℝ\)
Vậy nên BPT luôn đúng với mọi số thực x
d, Ta có :\(-x^2+3x-3=-\left(x^2-3x+3\right)\)
\(=-\left(x^2-2.\frac{3}{2}.x+\frac{9}{4}\right)-\frac{3}{4}=-\left(x-\frac{3}{2}\right)^2-\frac{3}{4}\)
Do \(\left(x-\frac{3}{2}\right)^2\ge0\forall x\inℝ\Leftrightarrow-\left(x-\frac{3}{2}\right)^2-\frac{3}{4}\le-\frac{3}{4}\forall x\inℝ\)
Hay \(-x^2+3x-3\le0\forall x\inℝ\)
Vậy nên BPT luôn đúng với mọi số thực x
2 câu còn lại bạn nào làm giúp mình nha
a) x4 + 6x3 + 11x2 + 6x + 1 = 0 <=> ( x2 + 3x + 1 ) 2 = 0 <=> x2 + 3x + 1 = 0
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\(1)\)
\(a)\)\(A=5-8x-x^2\)
\(A=-\left(x^2+8x+16\right)+21\)
\(A=-\left(x+4\right)^2+21\le21\)
Dấu "=" xảy ra \(\Leftrightarrow\)\(-\left(x+4\right)^2=0\)
\(\Leftrightarrow\)\(x=-4\)
Vậy GTLN của \(A\) là \(21\) khi \(x=-4\)
\(b)\)\(B=5-x^2+2x-4y^2-4y\)
\(-B=\left(x^2-2x+1\right)+\left(4y^2+4y+1\right)-7\)
\(-B=\left(x-1\right)^2+\left(2y+1\right)^2-7\ge-7\)
\(B=-\left(x-1\right)^2-\left(2y+1\right)^2+7\le7\)
Dấu "=" xảy ra \(\Leftrightarrow\)\(\hept{\begin{cases}-\left(x-1\right)^2=0\\-\left(2y+1\right)^2=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=1\\y=\frac{-1}{2}\end{cases}}}\)
Vậy GTLN của \(B\) là \(7\) khi \(x=1\) và \(y=\frac{-1}{2}\)
Chúc bạn học tốt ~
\(2)\)\(A=\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)
\(2A=2\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)
\(2A=\left(3-1\right)\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)
\(2A=\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)
\(2A=\left(3^4-1\right)\left(3^4+1\right).....\left(3^{64}+1\right)\)
\(............\)
\(2A=\left(3^{64}-1\right)\left(3^{64}+1\right)\)
\(2A=3^{128}-1\)
\(A=\frac{2^{128}-1}{3}\)
Chúc bạn học tốt ~
1. \(x^4+6x^3+11x^2+6x+1=0\)
\(\Leftrightarrow x^4+6x^3+9x^2+2x^2+6x+1=0\)
\(\Leftrightarrow\left(x^2+3x+1\right)^2=0\)
\(\Leftrightarrow x^2+3x+1=0\)
\(\Leftrightarrow\left(x+\frac{3}{2}\right)^2-\frac{5}{4}=0\)
\(\Leftrightarrow\left(x+\frac{3}{2}\right)^2=\frac{5}{4}\)
\(\Leftrightarrow\orbr{\begin{cases}x+\frac{3}{2}=\frac{\sqrt{5}}{2}\\x+\frac{3}{2}=-\frac{\sqrt{5}}{2}\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{-3+\sqrt{5}}{2}\\x=-\frac{3+\sqrt{5}}{2}\end{cases}}\)
2. \(x^4+x^3-4x^2+x+1=0\)
\(\Leftrightarrow\left(x^4+2x^2+1\right)+2.\frac{x}{2}\left(x^2+1\right)+\left(\frac{x}{2}\right)^2-\left(\frac{5}{2}x\right)^2=0\)
\(\Leftrightarrow\left(x^2+1+\frac{x}{2}\right)^2-\left(\frac{5}{2}x\right)^2=0\)
\(\Leftrightarrow\left(x^2-1\right)^2\left(x^2+3x+1\right)=0\)
\(\Leftrightarrow\hept{\begin{cases}\left(x-1\right)^2=0\\x^2+3x+1=0\end{cases}}\)
+) ( x - 1 )2 = 0
<=> x - 1 = 0
<=> x = 1
+) x2 + 3x + 1 = 0
<=> ( x + 3/2 )2 - 5/4 = 0
<=> ( x + 3/2 )2 = 5/4
<=> \(\hept{\begin{cases}x+\frac{3}{2}=\frac{\sqrt{5}}{2}\\x+\frac{3}{2}=-\frac{\sqrt{5}}{2}\end{cases}}\)
<=> \(\hept{\begin{cases}x=\frac{-3+\sqrt{5}}{2}\\x=-\frac{3+\sqrt{5}}{2}\end{cases}}\)
Vậy pt có tập nghiệm \(S=\left\{1;\frac{-3+\sqrt{5}}{2};-\frac{3+\sqrt{5}}{2}\right\}\)
1) \(2x^4+5x^2+2=0\)
\(\Leftrightarrow\left(2x^2+1\right)\left(x^2+2\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}2x^2+1=0\\x^2+2=0\end{cases}}\Rightarrow\orbr{\begin{cases}x^2=-\frac{1}{2}\\x^2=-2\end{cases}}\) (vô lý)
=> pt vô nghiệm
2) \(2x^4-7x^2-4=0\)
\(\Leftrightarrow\left(x^2-4\right)\left(2x^2+1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x^2-4=0\\2x^2+1=0\end{cases}\Leftrightarrow}\orbr{\begin{cases}x^2=4\\x^2=-\frac{1}{2}\left(vl\right)\end{cases}\Rightarrow}\orbr{\begin{cases}x=2\\x=-2\end{cases}}\)
3) \(x^4-5x^2+4=0\)
\(\Leftrightarrow\left(x^2-1\right)\left(x^2-4\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x^2-1=0\\x^2-4=0\end{cases}\Leftrightarrow}\orbr{\begin{cases}x^2=1\\x^2=4\end{cases}\Rightarrow}\orbr{\begin{cases}x=\pm1\\x=\pm2\end{cases}}\)
4) \(2x^4-20x^2+18=0\)
\(\Leftrightarrow x^4-10x^2+9=0\)
\(\Leftrightarrow\left(x^2-1\right)\left(x^2-9\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x^2-1=0\\x^2-9=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x^2=1\\x^2=9\end{cases}\Rightarrow}\orbr{\begin{cases}x=\pm1\\x=\pm3\end{cases}}\)
1. \(2x^4+5x^2+2=0\)
Vì \(2x^4+5x^2+2\ge2\)
=> Pt trên vô nghiệm
2. \(2x^4-7x^2-4=0\)
\(\Leftrightarrow2x^4+x^2-8x^2-4=0\)
\(\Leftrightarrow x^2\left(2x^2+1\right)-4\left(2x^2+1\right)=0\)
\(\Leftrightarrow\left(x^2-4\right)\left(2x^2+1\right)=0\)
\(\Leftrightarrow\left(2x^2+1\right)\left(x+2\right)\left(x-2\right)=0\)
\(\Leftrightarrow\hept{\begin{cases}2x^2+1=0\left(vo-ly\right)\\x+2=0\\x-2=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=-2\\x=2\end{cases}}\)