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

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

\(=-\left[\left(x-1\right)^2-4\right]=-\left(x-1\right)^2+4\le4\)

Vậy \(A_{max}=4\Leftrightarrow x-1=0\Leftrightarrow x=1\)

\(B=-2x^2-4x=-2\left(x^2+2x\right)\)

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

\(=-2\left[\left(x+1\right)^2-1\right]=-\left(x+1\right)^2+2\le2\)

Vậy \(B_{max}=2\Leftrightarrow x+1=0\Leftrightarrow x=-1\)

\(C=-x^2-6x+12=-\left(x^2+6x-12\right)\)

\(=-\left(x^2+6x+9-21\right)\)

\(=-\left[\left(x+3\right)^2-21\right]=-\left(x+3\right)^2+21\le21\)

Vậy \(C_{max}=21\Leftrightarrow x+3=0\Leftrightarrow x=-3\)

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

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

\(=-\left[\left(x-\frac{3}{2}\right)^2-\frac{5}{4}\right]=-\left(x-\frac{3}{2}\right)^2+\frac{5}{4}\le\frac{5}{4}\)

Vậy \(D_{max}=\frac{5}{4}\Leftrightarrow x-\frac{3}{2}=0\Leftrightarrow x=\frac{3}{2}\)

Bài làm

a) x² - 5x - 14

= x² + 2x - 7x - 14

= ( x² + 2x ) - ( 7x + 14 )

= x( x + 2 ) - 7( x + 2 )

= ( x + 2 )( x - 7 )

# Học tốt #

2^{2n + 3} - 4^{n + 1} - 2^{2n + 1} = 160

=> 2²ⁿ.8 - 2^{2n + 2} - 2²ⁿ.2 = 160

=> 2²ⁿ.8 - 2²ⁿ.4 - 2²ⁿ.2 = 160

=> 2²ⁿ(8 - 4 - 2) = 160

=> 2²ⁿ.2 = 160

=> 2²ⁿ = 160 : 2

=> 2²ⁿ = 80

(xem lại đề)

Phân tích :

x²  - 5x - 14 = x² - 7x + 2x - 14 = x(X - 7) + 2(x - 7) = (x + 2)(x - 7)

x² - xy - 12y² = x² - 4xy + 3xy - 12y² = x(x - 4y) + 3y(x - 4y) = (x + 3y)(x - 4y)

a) \(x^2-9=2\left(x+3\right)^2\)

\(\Leftrightarrow x^2-9=2x^2+12x+18\)

\(\Leftrightarrow x^2-2x^2-12x=18+9\)

\(\Leftrightarrow-x^2-12x=27\)

\(\Leftrightarrow x^2+12x+27=0\)

\(\Leftrightarrow\left(x+6\right)^2=9=3^2=\left(-3\right)^2\)

\(\Leftrightarrow\orbr{\begin{cases}x+6=3\\x+6=-3\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x=-3\\x=-9\end{cases}}\)

\(E=2x^2+y^2-2xy-8x+24\)

\(=x^2+x^2+y^2-2xy-8x+16+8\)

\(=\left(x^2-2xy+y^2\right)+\left(x^2-8x+16\right)+8\)

\(=\left(x-y\right)^2+\left(x-4\right)^2+8\)

Vậy \(E_{min}=8\Leftrightarrow x=y=4\)

Bài làm

E = 2x² + y² - 2xy - 8x + 24

E = ( x² - 2xy + y² ) + ( x² - 8x + 16 ) + 8

E = ( x² - 2xy + y² ) + ( x² - 2.4x + 4² ) + 8

E = ( x - y )² + ( x - 4 )² + 8 > 8

Dấu " = " xảy ra <=> E = 8

                          <=> x = 4; y = 4

Vậy E nhận giá trị nhỏ nhất là 8 khi x = 4 và y = 4

# Học tốt #

Giúp tuii với huhuu

\(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-z^2x^2\left(z-x\right)\)

\(=x^2y^2\left(y-x\right)-y^2z^2\left[\left(y-x\right)-\left(z-x\right)\right]-z^2x^2\left(z-x\right)\)

\(=x^2y^2\left(y-x\right)-y^2z^2\left(y-x\right)+y^2z^2\left(z-x\right)-z^2x^2\left(z-x\right)\)

\(=y^2\left(y-x\right)\left(x-z\right)\left(x+z\right)-z^2\left(x-z\right)\left(y-x\right)\left(y+x\right)\)

\(=\left(y-x\right)\left(x-z\right)\left(xy^2+y^2z-z^2y-z^2x\right)\)

Xet \(xy^2+y^2z-z^2y-z^2x=x\left(y-z\right)\left(y+z\right)+yz\left(y-z\right)=\left(y-z\right)\left(xy+yz+zx\right)\)

Vay \(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-z^2x^2\left(z-x\right)=\left(y-x\right)\left(x-z\right)\left(y-z\right)\left(xy+yz+zx\right)\)

\(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-z^2x^2\left(z-x\right)\)

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

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

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

\(=\left(x-z\right)\left(xy^3+y^3z-y^2x^2-y^2xz-y^2z^2+z^2x^2\right)\)

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