Bài 8
a, 25x^2 - 9 =0
<=> (5x-9)(5x+9) = 0
<=> 5x-9=0
       5x+9=0
<=> 5x=9
       5x=-9
<=> x=9/5
       x=-9/5
b, (x+4)^2 - (x+1)(x-1) = 16
    (x+4)^2 - 16 - (x^2 - 1) = 0
    (x+4-4)(x+4+4) - x^2 + 1 =0
    x(x+8) - x^2 + 1 = 0
    x^2 + 8x - x^2 + 1 = 0
       8x = -1
        x = -1/8

Bài 7:
A = 9x^2 + 42x + 49
   = (3x+7)^2
Vói x=1 ta có :
A = (3.1+7)^2
   = 10^2 = 100
B= 25x^2 - 2xy + 1/25.y^2
  = (5x-1/5y)^2
Với x=-1/5 và y = -5 ta có:
B = [5.(-1/5) - 1/5(-5)]^2
    = (-1+1)^2 = 0
C = x^2 + 10x + 26
   = (x+5)^2 + 1
Với x=95 ta có:
C = (95+5)^2 + 1
   = 100^2 + 1
   = 1001

Bài 8
a, 25x^2 - 9 =0
<=> (5x-3)(5x+3) = 0
<=> 5x-3=0
       5x+3=0
<=> 5x=3
       5x=-3
<=> x=3/5
       x=-3/5
b, (x+4)^2 - (x+1)(x-1) = 16
    (x+4)^2 - 16 - (x^2 - 1) = 0
    (x+4-4)(x+4+4) - x^2 + 1 =0
    x(x+8) - x^2 + 1 = 0
    x^2 + 8x - x^2 + 1 = 0
       8x = -1
        x = -1/8

Bài 9
a, GTLN:
A = -x^2 + 2x + 3
   = -(x^2 - 2x + 1) + 4
   = -(x-1)^2 + 4
Vì -(x-1)^2 ≤ 0
=> A = -(x-1)^2 + 4 ≤ 4
Dấu "=" xảy ra <=> x-1=0
                       <=> x = 1
Vậy Max A = 4 <=> x=1
B = -x^2 - 4x + 1
   = -(x^2 + 4x + 4) + 5
   = -(x+2)^2 + 5
Vì -(x+2)^2 ≤ 0
=> B = -(x+2)^2 + 5 ≤ 5
Dấu "=" xảy ra <=> x+2=0
                      <=> x=-2
Vậy Max B = 5 <=> x=-2
C = 5 - 8x - x^2
   = -(x^2 + 8x + 16) + 21
   = -(x+4)^2 + 21
Vì -(x+4)^2 ≤ 0
=> C = -(x+4)^2 ≤ 21
Dấu "=" xảy ra <=> x+4=0
                      <=> x = -4
Vậy Max C = 21 <=> x=-4

Bài 9
b, GTNN
B= 4x^2 + 6x + 5
  = 4x^2 + 6x + 9/4 + 11/4
  = (2x+3/2)^2 + 11/4
Vì (2x+3/2)^2 ≥ 0
=> N = (2x+3/2)^2 + 11/4 ≥ 11/4
Dấu "=" xảy ra <=> 2x+3/2=0
                          <=> 2x = -3/2
                          <=> x = -3/4
Vậy MinB = 11/4 <=> x=-3/4
A = 4x^2 + 4x + 11
   = 4x^2 + 4x + 1 + 10
   = (2x+1)^2 + 10
Vì (2x+1)^2 ≥ 0
=> A = (2x+1)^2 + 10 ≥ 10
Dấu "=" xảy ra <=> 2x+1=0
                          <=> 2x = -1
                          <=> x = -1/2
Vậy Min A = 10 <=> x=-1/2

Bài 11
Có (5a-3b+4c)(5a-3b-4c) = (3a-5b)^2
=> (5a-3b)^2 - (4c)^2 = (3a-5b)^2
=> 25a^2 - 30ab + 9b^2 - 16c^2 = 9a^2 - 30ab + 25b^2
=> 25a^2 - 30ab + 9b^2 - 16c^2 - 9a^2 + 30ab - 25b^2 = 0
=> 16a^2 - 16b^2 - 16c^2 = 0
=> 16(a^2 - b^2 - c^2) = 0
=> a^2 - b^2 - c^2 = 0
=> a^2 = b^2 + c^2
=> a; b; c là độ dài 3 cạnh của 1 tam giác vuông

Bài 10:
B = x^4 + y^4
   = x^4 + 2x^2.y^2 + y^4 - 2x^2.y^2
   = (x^2 + y^2)^2 - 2(xy)^2
Có x^2 + y^2 = 45 (cmt) và xy=18
=> B = 45^2 - 2.18^2
        = 2025 - 2.324
        = 2025 - 648
        = 1377
 

Bài 9
a, GTLN
D = -x^2 - x + 1
= -(x^2 + x + 1/4) + 5/4
= -(x+1/2)^2 + 5/4 ≤ 5/4
Dấu "=" xảy ra <=> x+1/2 = 0
<=> x = -1/2
Vậy Max D = 5/4 <=> x = -1/2

E = 5 - x^2 + 2x - 4y^2 - 4y
= -(x^2 - 2x + 1) - (4y^2 + 4y + 1) + 7
= -(x-1)^2 - (2y+1)^2 + 7 ≤ 7
Dấu "=" xảy ra <=> x-1 = 0
2y+1 = 0
<=> x = 1
y = -1/2
Vậy Max E = 7 <=> x = 1; y = -1/2

F = 3 - 10x^2 - 4xy - 4y^2
= 3 - 9x^2 - (x^2 + 4xy + 4y^2)
= 3 - 9x^2 - (x+2y)^2 ≤ 3
Dấu "=" xảy ra <=> 9x^2 = 0
x+2y = 0
<=> x^2 = 0
<=> x = -2y
<=> x = y = 0
Vậy Max F = 3 <=> x = y = 0

Bài 9 b, GTNN
C = 3x^2 - 6x - 1
   = 3(x^2 - 2x + 1) - 4
   = 3(x-1)^2 - 4 ≥ -4
Dấu "=" xảy ra <=> x-1 = 0
                      <=> x = 1
Vậy Min C = -4 <=> x = 1

Bài 8
l, x^2 + 2y^2 + z^2 - 2xy - 2y - 4z + 5 = 0
<=> (x^2 - 2xy + y^2) + (y^2 - 2y + 1) + (z^2 - 4z + 4) = 0
<=> (x-y)^2 + (y-1)^2 + (z-2)^2 = 0
<=> x-y = 0
       y-1 = 0
       z-2 = 0
<=> x = y
       y = 1
       z = 2
<=> x = y = 1
       z = 2

Bài 8
c, (2x-1)^2 + (x+3)^2 - 5(x+7)(x-7) = 0
<=> 4x^2 - 4x + 1 + x^2 + 6x + 9 - 5x^2 + 245 = 0
<=> 2x + 255 = 0
<=> 2x = -255
,<=> x = -255/2

d, (5x-1)^2 - (5x-4)(5x+4) = 7
<=> 25x^2 - 10x + 1 - (25x^2 - 16) = 7
<=> 25x^2 - 10x + 1 - 25x^2 + 16 = 7
<=> -10x + 17 = 7
<=> -10x = 7 - 17
<=> -10x = -10
<=> x = 1

bài 8
e, (4x-1)^2 - (2x+3)^2 + 5(x+2)^2 + 3(x-2)(x+2) = 500
<=> (4x-1-2x-3)(4x-1+2x+3) + 5(x^2 + 4x + 4) + 3(x^2 - 4) = 500
<=> (2x-4)(6x+2) + 5x^2 + 20x + 20 + 3x^2 - 12 = 500
<=> 12x^2 - 20x - 8 + 8x^2 + 20x + 8 = 500
<=> 20x^2 = 500
<=> x^2 = 25
<=> x = ±5
f, x^2 + y^2 + z^2 = 2x + 2y + 2z - 3
<=> x^2 - 2x + 1 + y^2 - 2y + 1 + z^2 - 2z + 1 = 0
<=> (x-1)^2 + (y-1)^2 + (z-1)^2 = 0
<=> x-1 = 0
y-1 = 0
z-1 = 0
<=> x = 1
y = 1
z = 1
<=> x = y = z = 1

Bài 8
i, y^2 + 2y + 4^x - 2^(x+1) + 2 = 0
<=> y^2 + 2y + 1 + 4^x - 2^x.2 + 1 = 0
<=> (y+1)^2 + (2^x - 1)^2 = 0
<=> y+1 = 0
       2^x - 1 = 0
<=> y = -1
       2^x = 1
<=> y = -1
       2^x = 2^0
<=> x = 0

k, 5x^2 + 5y^2 + 8xy - 2x + 2y + 2 = 0 
<=> (4x^2 + 8xy + 4y^2) + (x^2 - 2x + 1) + (y^2 + 2y + 1) = 0
<=> (2x+2y)^2 + (x-1)^2 + (y+1)^2 = 0
<=> 2x+2y = 0
       x-1 = 0
       y+1 = 0
<=> x = -y
       x = 1
       y = -1
<=> x = 1
       y = -1