Question

[Ngọc Nguyễn Ánh]

1/ -chứng minh rằng: x^2 -6x+10>0 với mọi x

- CMR: x^2 -2xy +y^2 +1 >0 với mọi x và y

2/ tìm giá trị nhỏ nhất của biểu thức M = x^2 -6x+12

3/Tìm x biết:

a/ ( x+3)^2 + (x-2)(x+2) - 2(x-1)^2=7

b) x^2+x=0

c) x^3 - 1/4 x=0

4/ Rút gọn biểu thức:

a) ( x+10)^2 - ( x^2 +2x)

b) ( x+2)(x-2) + (x-1)(x^2 + x+1) - x(x^2 +x)

Answer 1

[Phương An]

\(A=x^2-6x+10\)

\(=x^2-6x+9+1\)

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

\(\left(x-3\right)^2\ge0\)

\(\Rightarrow\left(x-3\right)^2+1\ge1>0\)

Vậy A > 0 với mọi x.

\(B=x^2-2xy+y^2+1\)

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

\(\left(x-y\right)^2\ge0\)

\(\Rightarrow\left(x-y\right)^2+1\ge1>0\)

Vậy B > 0 với mọi x, y.

\(M=x^2-6x+12\)

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

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

\(\left(x-3\right)^2\ge0\)

\(\Rightarrow\left(x-3\right)^2+3\ge3\)

\(MinB=3\Leftrightarrow x=3\)

\(\left(x+3\right)^2+\left(x-2\right)\left(x+2\right)-2\left(x-1\right)^2=7\)

\(x^2+6x+9+x^2-4-2\left(x^2-2x+1\right)=7\)

\(2x^2+6x+5-2x^2+4x-2=7\)

\(10x=7+3\)

\(10x=10\)

\(x=1\)

\(x^2+x=0\)

\(x\left(x+1\right)=0\)

\(\left[\begin{array}{nghiempt}x=0\\x+1=0\end{array}\right.\)

\(\left[\begin{array}{nghiempt}x=0\\x=-1\end{array}\right.\)

\(x^3-\frac{1}{4}x=0\)

\(x\left(x^2-\frac{1}{4}\right)=0\)

\(x\left(x-\frac{1}{2}\right)\left(x+\frac{1}{2}\right)=0\)

\(\left[\begin{array}{nghiempt}x=0\\x-\frac{1}{2}=0\\x+\frac{1}{2}=0\end{array}\right.\)

\(\left[\begin{array}{nghiempt}x=0\\x=\frac{1}{2}\\x=-\frac{1}{2}\end{array}\right.\)

\(\left(x+10\right)^2-\left(x^2+2x\right)\)

\(=x^2+20x+100-x^2-2x\)

\(=18x+100\)

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

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

\(=-5\)