A = 5 - 8\(x^2\)

Vì \(x^2\) ≥ 0 ∀ \(x\) ∈ R

⇒ -8\(x^2\) ≤ 0

⇒ A = 5 - 8\(x^2\) ≤ 5 dấu bằng xảy ra khi \(x=0\)

Vậy Amax = 5 khi \(x\) = 0

\(x^2\ge0\forall x\)

=>\(-8x^2\le0\forall x\)

=>\(-8x^2+5\le5\forall x\)

Dấu '=' xảy ra khi x=0

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Ta có: EF//BC

AH⊥BC

Do đó: AH⊥FE tại E

=>ΔAEF vuông tại E

Xét tứ giác BEKA có \(\hat{BEK}+\hat{BAK}=90^0+90^0=180^0\)

nên BEKA là tứ giác nội tiếp

=>\(\hat{EBK}=\hat{EAK}\)

=>\(\hat{EBK}=\hat{HAC}\)

mà \(\hat{HAC}=\hat{HBA}\left(=90^0-\hat{HAB}\right)\)

và \(\hat{HBA}=\hat{AFE}\) (hai góc đồng vị, CB//EF)

nên \(\hat{EBK}=\hat{AFE}\)

Xét ΔEBK vuông tại E và ΔEFA vuông tại E có

\(\hat{EBK}=\hat{EFA}\)

Do đó: ΔEBK~ΔEFA

=>\(\frac{BK}{FA}=\frac{BE}{FE}\)

=>\(BK\cdot FE=BE\cdot FA\)

ê mình hỏi

ab^3 hay là (ab)^3 thế

6A:

a: \(\frac{3}{x^2-3x}=\frac{3}{x\left(x-3\right)}=\frac{3\cdot2}{2x\left(x-3\right)}=\frac{6}{2x\left(x-3\right)}\)

\(\frac{5}{2x-6}=\frac{5}{2\left(x-3\right)}=\frac{5\cdot x}{2\left(x-3\right)\cdot x}=\frac{5x}{2x\left(x-3\right)}\)

b: \(\frac{3}{x^2-4}=\frac{3}{\left(x-2\right)\left(x+2\right)}=\frac{3\cdot\left(x-2\right)}{\left(x-2\right)\left(x-2\right)\left(x+2\right)}=\frac{3x-6}{\left(x-2\right)^2\cdot\left(x+2\right)}\)

\(\frac{x}{x^2-4x+4}=\frac{x}{\left(x-2\right)^2}=\frac{x\cdot\left(x+2\right)}{\left(x-2\right)^2\cdot\left(x+2\right)}\)

6B:

a: \(\frac{5x}{2x+8}=\frac{5x}{2\left(x+4\right)}=\frac{5x\cdot3}{2\cdot3\cdot\left(x+4\right)}=\frac{15x}{6\left(x+4\right)}\)

\(\frac{x+2}{3x+12}=\frac{x+2}{3\left(x+4\right)}=\frac{\left(x+2\right)\cdot2}{3\cdot\left(x+4\right)\cdot2}=\frac{2x+4}{6\left(x+4\right)}\)

b: \(\frac{7}{x^2-6x+9}=\frac{7}{\left(x-3\right)^2}=\frac{7\cdot3x}{3x\left(x-3\right)^2}=\frac{21x}{3x\left(x-3\right)^2}\)

\(\frac{x}{3x^2-9x}=\frac{x}{3x\left(x-3\right)}=\frac{x\left(x-3\right)}{3x\left(x-3\right)\left(x-3\right)}=\frac{x^2-3x}{3x\left(x-3\right)^2}\)

7A:

a: \(\frac{10}{x+3}=\frac{10\cdot2\cdot\left(x-3\right)}{2\left(x+3\right)\left(x-3\right)}=\frac{20x-60}{2\left(x+3\right)\left(x-3\right)}\)

\(\frac{5}{2x-6}=\frac{5}{2\left(x-3\right)}=\frac{5\cdot\left(x+3\right)}{2\left(x-3\right)\left(x+3\right)}=\frac{5x+15}{2\left(x-3\right)\left(x+3\right)}\)

\(\frac{-1}{x^2-9}=\frac{-1}{\left(x-3\right)\left(x+3\right)}=\frac{-1\cdot2}{2\cdot\left(x-3\right)\left(x+3\right)}=-\frac{2}{2\left(x-3\right)\left(x+3\right)}\)

b: \(\frac{1}{2x-y}=\frac{4\left(x-y\right)^2}{4\left(2x-y\right)\left(x-y\right)^2}=\frac{4x^2-8xy+4y^2}{4\left(2x-y\right)\left(x-y\right)^2}\)

\(\frac{x}{4x-4y}=\frac{x}{4\left(x-y\right)}=\frac{x\left(x-y\right)\left(2x-y\right)}{4\left(x-y\right)\left(x-y\right)\left(2x-y\right)}=\frac{\left(x^2-xy\right)\left(2x-y\right)}{4\left(x-y\right)^2\cdot\left(2x-y\right)}\)

\(\frac{-1}{x^2-2xy+y^2}=\frac{-1}{\left(x-y\right)^2}=\frac{-1\cdot4\cdot\left(2x-y\right)}{4\left(2x-y\right)\left(x-y\right)^2}=\frac{-8x+4y}{4\left(2x-y\right)\left(x-y\right)^2}\)

7B:

a: \(\frac{-7}{x-4}=\frac{-7\cdot3\cdot\left(x+4\right)}{\left(x-4\right)\left(x+4\right)\cdot3}=\frac{-21x-84}{3\left(x-4\right)\left(x+4\right)}\)

\(\frac{3}{3x+12}=\frac{3}{3\left(x+4\right)}=\frac{3\left(x-4\right)}{3\left(x+4\right)\cdot\left(x-4\right)}=\frac{3x-12}{3\left(x+4\right)\left(x-4\right)}\)

\(\frac{-5}{16-x^2}=\frac{5}{x^2-16}=\frac{5}{\left(x-4\right)\left(x+4\right)}=\frac{5\cdot3}{3\left(x-4\right)\left(x+4\right)}=\frac{15}{3\left(x-4\right)\left(x+4\right)}\)

b: \(\frac{1}{2x-y}=\frac{1\cdot\left(2x-y\right)\left(2x+y\right)}{\left(2x-y\right)\left(2x-y\right)\left(2x+y\right)}=\frac{4x^2-y^2}{\left(2x-y\right)^2\cdot\left(2x+y\right)}\)

\(\frac{-2}{4x^2-y^2}=\frac{-2}{\left(2x-y\right)\left(2x+y\right)}=\frac{-2\cdot\left(2x-y\right)}{\left(2x-y\right)\left(2x+y\right)\left(2x-y\right)}=\frac{-4x+2y}{\left(2x-y\right)^2\cdot\left(2x+y\right)}\)

\(\frac{2x^2+y^2}{4x^2-4xy+y^2}=\frac{2x^2+y^2}{\left(2x-y\right)^2}=\frac{\left(2x^2+y^2\right)\left(2x+y\right)}{\left(2x-y\right)^2\cdot\left(2x+y\right)}\)

Câu 3a:

4\(x^3\) - 9\(x\)

= \(x\) x (4\(x^2\) - 9)

= \(x\) x [(2\(x\))\(^2\) - 3\(^2\)]

= \(x\times\) [2\(x\) - 3][\(2x+3\)]

b; \(x^2+2x-3\)

= \(x^2-x+3x-3\)

= \(\left(x^2-x\right)+\left(3x-3\right)\)

= \(x\left(x-1\right)+3\left(x-1\right)\)

= (\(x-1)\)(\(x+3\))

Câu c:

\(x^2\) - y\(^2\) - 6\(x\) + 9

= (\(x^2\) - 6\(x\) + 9) - y\(^2\)

= (\(x^2-2.3x\) + 3\(^2\)) - y\(^2\)

= (\(x-3\))\(^2\) - y\(^2\)

= (\(x-3-y\))(\(\)\(x-3+y\))

Sửa đề: \(2a^2+7ab+3b^2=0\)

=>\(2a^2+6ab+ab+3b^2=0\)

=>2a(a+3b)+b(a+3b)=0

=>(a+3b)(2a+b)=0

=>\(\left[\begin{array}{l}a+3b=0\\ 2a+b=0\end{array}\right.\Rightarrow\left[\begin{array}{l}a=-3b\\ b=-2a\end{array}\right.\)

TH1: a=-3b

\(\frac{8a-3b}{2a-b}-\frac{2a-5b}{2a+b}\)

\(=\frac{8\cdot\left(-3b\right)-3b}{2\left(-3b\right)-b}-\frac{2\cdot\left(-3b\right)-5b}{2\cdot\left(-3b\right)+b}=\frac{-24b-3b}{-6b-b}-\frac{-6b-5b}{-6b+b}\)

\(=\frac{-27}{-7}-\frac{-11}{-5}=\frac{27}{7}-\frac{11}{5}=\frac{135}{35}-\frac{77}{35}=\frac{58}{35}\)

TH2: b=-2a

\(\frac{8a-3b}{2a-b}-\frac{2a-5b}{2a+b}\)

\(=\frac{8a-3\cdot\left(-2a\right)}{2a-\left(-2a\right)}-\frac{2a-5\cdot\left(-2a\right)}{2a-2a}=\frac{14a}{4a}-\frac{12a}{0a}\)

=>Khi b=-2a thì biểu thức không có giá trị

Ta có: \(\left(2x-1\right)\left(4x^2+2x+1\right)-2x\left(2x-3\right)\left(2x+3\right)=81x^2\)

=>\(8x^3-1-2x\left(4x^2-9\right)=81x^2\)

=>\(8x^3-1-8x^3+18x=81x^2\)

=>\(81x^2=18x-1\)

=>\(81x^2-18x+1=0\)

=>\(\left(9x-1\right)^2=0\)

=>9x-1=0

=>9x=1

=>\(x=\frac19\)

\(\left(2x-1\right)\left(4x^2+2x+1\right)-2x\left(2x-3\right)\left(2x+3\right)=81x^2\)

\(8x^3-1-2x\left(4x^2-9\right)=81x^2\)

\(8x^3-1-\left(8x^3-18x\right)=81x^2\)

\(8x^3-1-8x^3+18x=81x^2\)

\(18x-1=81x^2\)

\(81x^2-18x+1=0\)

\(\left(9x\right)^2-2\cdot9x+1=0\)

\(\left(9x-1\right)^2=0\)

\(9x-1=0\)

\(x=\frac19\)

Vậy \(x=\frac19\)