1) Tính nhanh
a)\(\frac{63^2-47^2}{215^2-105^2}\)
b)\(\frac{2006^3+1}{2006^2-2005}\)
c)\(\frac{2006^3-1}{2006^2+2007}\)
d)10\(^2\)+8\(^2\)+6\(^2\)+......+2\(^2\)-(9\(^2\)+7\(^2\)+......+1\(^2\))
2)Tìm x, pk
a)4(x+1)\(^2\)+(2x-1)\(^2\)-8(x-1)(x+1)=11
b)(x-3)(x\(^2\)+3x+9)+x(x+2)(2-x)=1
c)(x+1)\(^3\)-(x-1)\(^3\)-6(x-1)\(^2\)=-10
Bài 2
\( a)4{\left( {x + 1} \right)^2} + {\left( {2x - 1} \right)^2} - 8\left( {x - 1} \right)\left( {x + 1} \right) = 11\\ \Leftrightarrow 4\left( {{x^2} + 2x + 1} \right) + 4{x^2} - 4x + 1 - 8\left( {{x^2} - 1} \right) = 11\\ \Leftrightarrow 4{x^2} + 8x + 4 + 4{x^2} - 4x + 1 - 8{x^2} + 8 = 11\\ \Leftrightarrow 4x + 13 = 11\\ \Leftrightarrow 4x = 11 - 13\\ \Leftrightarrow 4x = - 2\\ \Leftrightarrow x = - \dfrac{1}{2} \)
Bài 2:
\( b)\left( {x - 3} \right)\left( {{x^2} + 3x + 9} \right) + x\left( {x + 2} \right)\left( {2 - x} \right) = 1\\ \Leftrightarrow {x^3} - 27 + x\left( {2 + x} \right)\left( {2 - x} \right) = 1\\ \Leftrightarrow {x^3} - 27 + x\left( {4 - {x^2}} \right) = 1\\ \Leftrightarrow {x^3} - 27 + 4x - {x^3} = 1\\ \Leftrightarrow 4x = 1 + 27\\ \Leftrightarrow 4x = 28\\ \Leftrightarrow x = 7 \)
Bài 2:
\( c){\left( {x + 1} \right)^3} - {\left( {x - 1} \right)^3} - 6{\left( {x - 1} \right)^2} = - 10\\ \Leftrightarrow {x^3} + 3{x^2} + 3x + 1 - {x^3} + 3{x^2} - 3x + 1 - 6{x^2} + 12x - 6 = - 10\\ \Leftrightarrow - 4 + 12x = - 10\\ \Leftrightarrow 12x = - 10 + 4\\ \Leftrightarrow 12x = - 6\\ \Leftrightarrow x = - \dfrac{1}{2} \)
\(a,\frac{63^2-47^2}{215^2-105^2}=\frac{\left(63-47\right)\left(63+47\right)}{\left(215-105\right)\left(215+105\right)}=\frac{16.110}{110.320}=\frac{16}{320}=\frac{1}{20}\)
\(b,\frac{2006^3+1}{2006^2-2005}=\frac{\left(2006+1\right)\left(2006^2-2006+1\right)}{2006^2-2005}=\frac{2007\left(2006^2-2005\right)}{2006^2-2005}=2007\)
\(c,10^2+8^2+.....+2^2-\left(9^2+7^2+....+1^2\right)=\left(10^2-9^2\right)+\left(8^2-7^2\right)+....+\left(2^2-1^2\right)=\left(10+9\right)\left(10-9\right)+\left(8+7\right)\left(8-7\right)+......+\left(2+1\right)\left(2-1\right)=\left(10+9+8+7+....+2+1\right)=\frac{10.9}{2}=45\)