[tex]k(x)=u(x)+v(x)=4x^2+9x-28\\m(x)=u(x)-v(x)=-2x^2-11x-12\\\\u(x)=4x^2+9x-28-v(x)\\4x^2+9x-28-v(x)-v(x)=-2x^2-11x-12\\-2v(x)=-2x^2-11x-12-4x^2-9x+28\\-2v(x)=-6x^2-20x+16\\-2v(x)=-2(3x^2+10x-8)\\\underline{v(x)=3x^2+10x-8}\\\\u(x)=4x^2+9x-28-(3x^2+10x-8)\\u(x)=4x^2+9x-28-3x^2-10x+8\\\underline{u(x)=x^2-x-20}[/tex]
Rozklad na czynniki pierwsze.
[tex]k(x)=4x^2+9x-28\\\Delta=9^2-4*4*(-28)=81+448\\\sqrt{\Delta}=\sqrt{529}=23\\x_1=\frac{-9-23}{8}=-4\\x_2=\frac{-9+23}8=\frac{14}8=\frac74\\k(x)=4(x-\frac74)(x+4)\\\underline{k(x)=(4x-7)(x+4)}[/tex]
[tex]m(x)=-2x^2-11x-12\\\Delta=(-11)^2-4*(-2)*(-12)=121-96\\\sqrt{\Delta}=\sqrt{25}=5\\x_1=\frac{11-5}{-4}=\frac{6}{-4}=-\frac64=-\frac32\\x_2=\frac{11+5}{-4}=\frac{16}{-4}=-4\\m(x)=-2(x+\frac32)(x+4)\\\underline{m(x)=-(2x+3)(x+4)}[/tex]
[tex]u(x)=x^2-x-20\\\Delta=(-1)^2-4*1*(-20)=1+80\\\sqrt{\Delta}=\sqrt{81}=9\\x_1=\frac{1-9}2=\frac{-8}2=-4\\x_2=\frac{1+9}2=\frac{10}2=5\\\underline{u(x)=(x+4)(x-5)}[/tex]
[tex]v(x)=3x^2+10x-8\\\Delta=10^2-4*3*(-8)=100+96\\\sqrt{\Delta}=\sqrt{196}=14\\x_1=\frac{-10-14}{6}=\frac{-24}6=-4\\x_2=\frac{-10+14}6=\frac{4}6=\frac23\\v(x)=3(x-\frac23)(x+4)\\\underline{v(x)=(3x-2)(x+4)}[/tex]
Odpowiedź:
układ równań
k(x) = u(x) + v(x) = 4x² + 9x - 28
m(x) = u(x) - v(x) = - 2x²- 11x - 12
dodajemy równania
u(x)+u(x) + v(x) - v(x) = 4x² - 2x²+ 9x - 11x - 28 - 12
2u(x) =2x²-2x -40
u(x)= (2/2)x² - (2/2)x - 40/2 = x²- x - 20
u(x) + v(x) = 4x²+ 9x - 28
x² - x - 20 + v(x) = 4x² + 9x - 28
v(x) = 4x² - x² + 9x + x - 28 + 20
v(x) = 3x² + 10x - 8
Przy rozłożeniu na czynniki obliczamy miejsca zerowe poszczególnych wielomianów.
1.
u(x) = x² - x - 20
x² - x - 20 = 0
a = 1 , b = - 1 , c = - 20
Δ = b² - 4ac = (- 1)² - 4 * 1 * (- 20) = 1 + 80 = 81
√Δ = √81 = 9
x₁ = (- b - √Δ)/2a = (1 - 9)/2 = - 8/2 = - 4
x₂ = ( - b + √Δ)/2a = (1 + 9)/2 = 10/2 = 5
Postać iloczynowa wielomianu = a(x - x₁)(x - x₂)
u(x) = (x + 4)(x - 5)
2.
v(x) = 3x² + 10x - 8
3x² + 10x - 8 = 0
a = 3 , b = 10 , c = - 8
Δ = b² - 4ac = 10² - 4 * 3 * (- 8) = 100 + 96 = 196
√Δ = √196 = 14
x₁ = (- b - √Δ)/2a = ( - 10 - 14)/6 = - 24/6 = - 4
x₂ = ( - b + √Δ)/2a = ( - 10 + 14)/6 = 4/6 = 2/3
v(x) = a(x - x₁)(x - x₂) = 3(x + 4)(x - 2/3)
3.
k(x) = 4x² + 9x - 28
4x² + 9x - 28 = 0
a = 4 , b = 9 , c = - 28
Δ = b² - 4ac = 9² - 4 * 4 * ( 28) = 81 + 448 = 529
√Δ= √529 = 23
x₁ = (- b - √Δ)/2a = ( - 9 - 23)/8 = - 32/8 = - 4
x₂ = ( - b + √Δ)/2a = (- 9 + 23)/8 = 14/9 = 1 5/9
k(x) = a(x- x₁)(x - x₂) = 4(x+ 4)(x - 1 5/9)
4.
m(x) = - 2x² - 11x - 12
-2x² - 11x - 12 = 0
a = - 2 , b = - 11 , c = - 12
Δ = b² - 4ac = (- 11)² - 4 * (- 2) * ( - 12) = 121 - 96 = 25
√Δ = √25 = 5
x₁ = ( - b - √Δ)/2a = ( 11 - 5)/(- 4) = 6/(- 4) = - 6/4 = - 3/2 = - 1 1/2
x₂ = ( - b + √Δ)/2a = (11 + 5)/(- 4) = 16/(- 4) = - 163/4 = - 4
m(x) = a(x - x₁)(x - x₂) = - 2(x + 1 1/2)(x + 4)
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