2.
б) x² +5x-6 = x² + 2 * (⁵/₂)x+ (⁵/₂)² - (⁷/₂)² = (x+ ⁵/₂)² - ⁴⁹/₄
в) 3x²-18x-15=3(x²-6x-5)=3(x²-2*3x+3² -14)=
=3((x-3)² -14)=3(x-3)² -42
3.
a) x²-4x+31=(x² -2*2x+4+27)=(x-2)² +27 >0 при любом х.
б) 9x²+24x+16=(3x+4)² >0 при любом х.
4.
а) -(х+2)² +5
х+2=0
х=-2
б) -х²+5х-4= -(х²-5х+4)= -(х² - 2 * ⁵/₂ х +(⁵/₂)² - ⁹/₄) =
= -((х - ⁵/₂)² - ⁹/₄) = -(х - ⁵/₂)² + ⁹/₄
х - ⁵/₂ =0
х= ⁵/₂
х=2,5
в) -3x²-9x+15=-3(x²+3x-5)=-3(x²+2*(³/₂) x+⁹/₄ - ²⁹/₄) =
= -3((x+³/₂)² - ²⁹/₄) = -3(x+³/₂)² + ⁸⁷/₄
x+³/₂=0
x= -³/₂
x= -1.5
5.
б) x²+7x+10= (x+5)(x+2)
x²+7x+10=0
D=7² - 4*10=9
x₁= -7-3 = -5
2
x₂ = -7+3 = -2
2
в) 5x²-7x+2=5(x-0.4)(x-1)=(5x-2)(x-1)
5x²-7x+2=0
D=(-7)² -4*5*2=49-40=9
x₁= 7-3 = 0.4
10
x₂ =7+3 =1
10
г) -2x²-9x-9= -2(x+3)(x+1.5)=(x+3)(-2x-3)
-2x²-9x-9=0
2x²+9x+9=0
D=9² -4*2*9=81-72=9
x₁= -9-3 = -3
4
x₂ = -9+3 = -1.5
4
6.
б) 2m²-3m-5 = 2(m+1)(m-2.5) = m+1 = m+1
4m²-12m+5 4(m-0.5)(m-2.5) 2(m-0.5) 2m-1
2m²-3m-5=0
D=(-3)² -4*2*(-5)=9+40=49
m₁= 3-7 = -1
4
m₂= 3+7 = 2.5
4
4m²-12m+5=0
D=(-12)² - 4*4*5 = 144 -80=64
m₁= 12-8 = 0.5
8
m₂ = 12+8 = 2.5
8
7*. y=x²-7x+12 = (x-3)(x-4) = x-3
x-4 x-4
x²-7x+12=0
D=(-7)² - 4*12=1
x₁= 7-1 = 3
2
x₂ = 7+1 =4
2
y=x-3 - прямая
х| 0 | 3
y| -3| 0
8*. 4kx²-5x+25k=0
D=(-5)² - 4*4k*25k=25 -400k <0<br>25 -400k <0<br>-400k< -25
400k > 25
k > 25/400
k > 1/16
Ответ: к> 1/16
