Question

Câu 1:

a, Phân tích đa thức thành nhân tử: 4(x² +15x +50)( x² +18x +72) - 3x²

b, Tìm các cặp số nguyên (x; y) thỏa mãn: y² +2(x² +1) = 2y(x+1)

Câu 2: Cho \(S=\dfrac{1}{51}+\dfrac{1}{52}+\dfrac{1}{53}+...+\dfrac{1}{99}+\dfrac{1}{100}\). So sánh S với \(\dfrac{1}{2}\)

Answer 1

\(4\left(x^2-15x+50\right)\left(x^2-18+72\right)-3x^2\)

\(=4\left(x+5\right)\left(x+10\right)\left(x+6\right)\left(x+12\right)-3x^2\)

\(=4\left[\left(x+5\right)\left(x+12\right)\right]\left[\left(x+10\right)\left(x+6\right)\right]-3x^2\)

\(=4\left(x^2+17x+60\right)\left(x^2+16x+60\right)-3x^2\)

Đặt \(x^2+16x+60=a\), ta có:

\(4\left(a+x\right)\left(a\right)-3x^2\)

\(=4a^2+4ax-3x^2\)

\(=4a^2-2ax+6ax-3x^2=2a\left(2a-x\right)+3x\left(2a-x\right)\)

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

Thay a vào ta có: \(\left[2\left(x^2+16x+60\right)-x\right]\left[2\left(x^2+16x+60\right)+3x\right]\)

\(=\left(2x^2+31x+120\right)\left(2x^2+35x+120\right)\)

Answer 2

Answer 3

S_1=1/1.2+1/3.4+1/5.6+...+1/99.100

=1‐1/2+1/3‐1/4+1/5‐1/6+...+1/99‐1/100

=﴾1+1/3+1/5+...+1/99﴿‐﴾1/2+1/4+1/6+...+1/100﴿

=﴾1+1/2+1/3+1/4+1/5+1/6+...+1/99+1/100﴿‐2﴾1/2+1/4+1/6+...+1/100﴿

=﴾1+1/2+1/3+1/4+...+1/100﴿‐﴾1+1/2+1/3+..+1/50﴿

=1/51+1/52+1/53+..+1/100 ﴾1﴿

S_2=1/51+1/52+1/53+..+1/100 ﴾2﴿

Từ ﴾1﴿,﴾2﴿=> S_1/S_2=1

Vậy S > 1/2