F4 Math一問

9) A ( x + 1 )^2 + B ( 2x - 1 )^2 = - 2x^2 + 8x + 1

A ( x^2 + 2x + 1 ) + B ( 4x^2 - 4x + 1 ) = - 2x^2 + 8x + 1

( A + 4B ) x^2 + ( 2A - 4B ) x + ( A + B ) = - 2x^2 + 8x + 1

So,

A + 4B = - 2 --- ( 1 )

2A - 4B = 8

A - 2B = 4 --- ( 2 )

( 1 ) - ( 2 ) : A + 4B - A + 2B = - 2 - 4

6B = -6

B = -1

A - 2 ( - 1 ) = 4 [ equ 2 ]

A = 2

So A = 2, B = -1.

11) A ( 2x + 1 ) + B ( 2x - 3 ) = 14x - 9

2Ax + A + 2Bx - 3B = 14x - 9

( 2A + 2B ) x + ( A - 3B ) = 14x - 9

So,

2A + 2B = 14

A + B = 7 --- ( 1 )

A - 3B = -9 --- ( 2 )

( 1 ) - ( 2 ) : A + B - A + 3B = 7 + 9

4B = 16

B = 4

A + 4 = 7 [ eqr 1 ]

A = 3

So A = 3, B = 4.

(13) 4x^2 + 5x - 2 = A ( x + 3 )( x - 1 ) + B ( 3x - 10 )

4x^2 + 5x - 2 = A ( x^2 + 2x - 3 ) + 3Bx - 10B

= Ax^2 + ( 2A + 3B ) x + ( - 3A - 10B )

So,

A = 4

2A + 3B = 5

2 ( 4 ) + 3B = 5

3B = - 3

B = -1

So A = 4, B = -1.

圖片參考：http://farm3.static.flickr.com/2417/2058248859_b4dce460bf.jpg?v=0


I am going to provide a faster method.
9)
A(x + 1)² + B(2x - 1)² = -2x² + 8x + 1
set x = 1/2,
9A/4 = 9/2 => A = 2,
set x = -1,
9B = -9 => B = -1,
A = 2, B = -1

11)
A(2x + 1) + B(2x - 3) = 14x - 9
set x = 3/2,
4A = 12 => A = 3
set x = -1/2,
-4B = -16 => B = 4
A = 3, B = 4


13)
4x² + 5x - 2 = A(x + 3)(x - 1) + B(3x - 10)
set x = 10/3,
(400/9) + (50/3) - 2 = A(19/3)(7/3)
550 - 18 = 133A
A = 4
set x = 1,
7 = -7B => B = -1
A = 4, B = -1