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Ta có:
\(\left(a-1\right)^2\ge0\Leftrightarrow a^2-2a+1\ge0\Leftrightarrow a^2+1\ge2a\) (1)
\(\left(b-1\right)^2\ge0\Leftrightarrow b^2-2b+1\ge0\Leftrightarrow b^2+1\ge2b\) (2)
\(\left(c-1\right)^2\ge0\Leftrightarrow c^2-2c+1\ge0\Leftrightarrow c^2+1\ge2c\) (3)
Từ (1), (2) và (3) suy ra;
\(a^2+1+b^2+1+c^2+1\ge2a+2b+2c\)
<=> \(a^2+b^2+c^2+3\ge2\left(a+b+c\right)\)
<=> \(a^2+b^2+c^2\ge2\left(a+b+c\right)-3\) \(\xrightarrow[]{}\) đpcm
Dấu "=" xảy ra khi a=b=c=1
Ta có: \(\left(a-1\right)^2+\left(b-1\right)^2+\left(c-1\right)^2\ge0\)
\(\Leftrightarrow a^2-2a+1+b^2-2b+1+c^2-2c+1\ge0\)
\(\Leftrightarrow a^2+b^2+c^2\ge+2a+2b+2c-3\)
\(\Leftrightarrow a^2+b^2+c^2\ge2\left(a+b+c\right)-3\) (đpcm)
Vậy \(a^2+b^2+c^2\ge2\left(a+b+c\right)-3\)
\(M=x^2+xy+y^2-3x-3y\)
\(\Rightarrow4M=4x^2+4xy+4y^2-12x-12y\)
\(=\left(x^2+4y^2+9+4xy-12y-6x\right)+\left(3x^2-6x+3\right)-12\)
\(=\left(x+2y-3\right)^2+3\left(x-1\right)^2-12\ge-12\)
\(\Rightarrow M\ge-3\)
\(\Rightarrow Min_M=-3\Leftrightarrow x=y=1\)
Ta có: A = \(x^2+xy+y^2-3x-3y\)
= \(\left(x^2+xy-3x\right)+y^2-3y\)
= \(\left[x^2+x\left(y-3\right)\right]+y^2-3y\)
= \(\left[x^2+2x\dfrac{\left(y-3\right)}{2}+\dfrac{\left(y-3\right)^2}{4}\right]-\dfrac{\left(y-3\right)^2}{4}+y^2-3y\)
= \(\left(x+\dfrac{y-3}{2}\right)^2-\dfrac{y^2-6y+9}{4}+\dfrac{4y^2}{4}-\dfrac{12y}{4}\)
= \(\left(x+\dfrac{y-3}{2}\right)^2+\dfrac{-y^2+6y-9+4y^2-12y}{4}\)
= \(\left(x+\dfrac{y-3}{2}\right)^2+\dfrac{3y^2-6y-9}{4}\)
= \(\left(x+\dfrac{y-3}{2}\right)^2+\dfrac{3\left(y^2-2y+1-4\right)}{4}\)
= \(\left(x+\dfrac{y-3}{2}\right)^2+\dfrac{3\left(y-1\right)^2-12}{4}\)
= \(\left(x+\dfrac{y-3}{2}\right)^2+\dfrac{3\left(y-1\right)^2}{4}-3\ge-3\)
Vậy Min A = -3 \(\Leftrightarrow\left\{{}\begin{matrix}y-1=0\\x+\dfrac{y-3}{2}=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=1\\x=1\end{matrix}\right.\)
Câu 4:
a: ĐKXĐ: \(x\notin\left\{0;-5\right\}\)
b: \(A=\dfrac{x^2+2x}{2\left(x+5\right)}+\dfrac{x-5}{x}+\dfrac{50-5x}{2x\left(x+5\right)}\)
\(=\dfrac{x^3+2x^2}{2x\left(x+5\right)}+\dfrac{2\left(x^2-25\right)}{2x\left(x+5\right)}+\dfrac{50-5x}{2x\left(x+5\right)}\)
\(=\dfrac{x^3+2x^2+2x^2-50+50-5x}{2x\left(x+5\right)}\)
\(=\dfrac{x^3+4x^2-5x}{2x\left(x+5\right)}=\dfrac{x\left(x^2+4x-5\right)}{2x\left(x+5\right)}\)
\(=\dfrac{x\left(x+5\right)\left(x-1\right)}{2x\left(x+5\right)}=\dfrac{x-1}{2}\)
c: Để A=-3 thì x-1=-6
hay x=-5(loại)
\(125-x^6=\left(5\right)^3-\left(x^2\right)^3\)
\(=\left(5-x^2\right)\left(25+5x^2+x^4\right)\)
\(49\left(x-4\right)^2-9\left(y+2\right)^2\)
\(=\left[7\left(x-4\right)\right]^2-\left[3\left(y+2\right)\right]^2\)
\(=\left[7x-28\right]^2-\left[3y+6\right]^2\)
\(=\left(7x-28-3y-6\right)\left(7x-28+3y+6\right)\)
\(=\left(7x-3y-34\right)\left(7x-22+3y\right)\)
Vẫn chưa hiểu dạng này hả em:)) Lần này chi tiết hết cỡ nhé
Xét \(\Delta BHA\) và \(\Delta CBA\) có:
\(\widehat{BHA}=\widehat{CBA}=90^o\)
\(\widehat{BAC}\) chung
\(⇒ Δ B H A ∼ Δ C B A ( g − g )\) (1)
Xét \(\Delta CHB\) và \(\Delta CBA\) có:
\(\widehat{CHB}=\widehat{CBA}=90^o\)
\(\widehat{BCA}\) chung
\(\Rightarrow\)\(Δ C HB ∼ Δ C BA(g-g)\) (2)
Từ (1) và (2) \(\Rightarrow\)\(Δ B H A ∼ Δ C H B ⇒ \) \(\dfrac{AH}{BH}=\dfrac{BH}{HC}\) \(⇒ B H ^2 = A H . H C\)
Tự vẽ hình và lưu ý ghi đủ đề bài ∆ABC vuông tại B e nhé