12 perfecting squares

X 2 + 6x – 7 = 0 X 2 + 6x = 7 ← pindah 7 ke kanan (x+3) 2 – 9 = 7 ← menyempurnakan kuasa dua (x+3) 2 = 7 + 9 ← pindah 9 ke kanan (x+3) 2 = 16 X+ 3 = 4 @ - 4 ← ambil punca kuasa dua X= 4 - 3 @ - 4 – 3 ← setiap satu tolak 3 X = 1 @ - 7

Kaedah menyempurnakan kuasa dua x 2 + 2 x + 1 = ( x+1) 2 -> x 2 + 4 x + 4 = ( x+2) 2 -> x 2 + 6 x + 9 = ( x+3) 2 -> x 2 + 8 x + 16 = ( x+4) 2 -> x 2 + 16 x + 64 = ( x+8) 2 -> x 2 + 2 x = ( x + 1 ) 2 – 1 x 2 + 4 x = ( x + 2 ) 2 – 4 x 2 + 6 x = ( x + 3 ) 2 – 9 x 2 + 8 x = ( x + 4 ) 2 – 16 x 2 + 16 x = ( x + 8 ) 2 - 64

Kaedah menyempurnakan kuasa dua x 2 - 2 x + 1 = ( x-1) 2 -> x 2 - 4 x + 4 = ( x-2) 2 -> x 2 - 6 x + 9 = ( x- 3) 2 -> x 2 - 8 x + 16 = ( x- 4) 2 -> x 2 - 16 x + 64 = ( x- 8) 2 -> x 2 - 2 x = ( x - 1 ) 2 – 1 x 2 - 4 x = ( x - 2 ) 2 – 4 x 2 - 6 x = ( x - 3 ) 2 – 9 x 2 - 8 x = ( x - 4 ) 2 – 16 x 2 - 16 x = ( x - 8 ) 2 - 64

Kaedah menyempurnakan kuasa dua

(x+4) 2 = 18 (x–3) 2 = 10 (x–2) 2 = 6 (x+ 1) 2 = 2 (x+4) 2 = 2+16 (x–3) 2 =1 + 9 (x–2) 2 = 2 +4 (x+1) 2 = 1 + 1 (x+4) 2 –16 = 2 (x–3) 2 – 9 = 1 (x–2) 2 – 4 = 2 (x+1) 2 – 1 = 1 X 2 + 8 x = 2 x 2 – 6 x = 1 x 2 – 4x = 2 x 2 + 2x = 1 X + 4 = ± √18 X – 3 = ±√10 X - 2 = ±√6 x + 1 = ±√2 x 2 + 8x-2=0 x 2 - 6x+1=0 x 2 - 4x-2=0 x 2 + 2x-1=0

X=0.243 @ - 7.243 X=5.828 @ 0.172 X = 4.449 @ - 0.449 X = 0.414 @ - 2.414 X= 4.243 - 4 @ - 4.243-4 X= 2.828 + 3 @ -2.828+3 X = 2.449 + 2 @ - 2.449+2 X = 1.414 - 1 @ - 1.414 – 1 X+4 = ± 4.243 X-3 = ± 2.828 X-2 = ± 2.449 X+1 = ± 1.414

(x-3.5) 2 = 11.25 (x+2.5) 2 = 9.25 (x–1.5) 2 = 0.25 (x+ 0.5) 2 = 1.25 X 2 -7x = -1 x 2 +5 x = 3 x 2 – 3x = -2 x 2 + x = 1 X + 4 = ± √11.25 X – 3 = ±√9.25 X – 1.5 = ±√0.25 x +0.5 = ±√1.25 x 2 -7x + 1=0 x 2 + 5x-3=0 x 2 - 3x+2=0 x 2 + x-1=0

X=6.854 @ 0.146 X=0.541 @ -5.541 X = 2 @ 1 X = 0.618 @ - 1.618 X= 3.354+3.5 @ - 3.354 + 3.5 X= 3.041-2.5 @ -3.041- 2.5 X = 0.5+1.5 @ - 0.5+1.5 X = 1.118-0.5 @ - 1.118 – 0.5 X-3.5 = ± 3.354 X+2.5 = ± 3.041 X-1.5 = ± 0.5 X+0.5 = ± 1.118

(x+2.5) 2 = 6.75 (x–1.5) 2 = 3 (x–2) 2 = 5.667 (x+ 1) 2 = 1.5 (x+2.5) 2 -6.25= 0.5 (x-1.5) 2 -2.25 =0.75 (x–2) 2 -4 = 1.667 (x+1) 2 -1= 0.5 X 2 + 5x = 0.5 X 2 – 3x = 0.75 X 2 – 4 x = 1.667 X 2 + 2x = 0.5 2X 2 + 10 x =1 4x 2 – 12 x = 3 3x 2 – 12x = 5 2x 2 + 4x = 1 X +2.5 = ± √6.75 X – 1.5 = ±√3 X - 2 = ±√5.667 x + 1 = ±√1.5 2x 2 +10x-1=0 4x 2 - 12x-3=0 3x 2 -12x-5=0 2x 2 +4X – 1 = 0

X=0.098 @ 5.098 X=3.232 @ -0.232 X = 4.381 @ - 0.381 X = 0.225 @ - 2.225 X= 2.598-2.5 @ - 2.598-2.5 X= 1.732+ 1.5 @ -1.732+1.5 X = 2.381 + 2 @ - 2.381+2 X = 1.225 - 1 @ - 1.225 - 1 X+2.5= ± 2.598 X-1.5 = ± 1.732 X-2 = ± 2.381 X+1 = ± 1.225

12 perfecting squares

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12 perfecting squares