ax^2+bx+c=0
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Zadanie 3
a)
x^2 - 10x + 25 = 0
\Delta =b^2-4ac=(-10)^2-4*1*25=100-100
\Delta=0, czyli x_1=x_2
x=\frac{-b}{2a}=\frac{10}{2*1}=5
b)
4$x^2$ = 12x - 9
4x^2-12x+9=0
\Delta = b^2-4ac=(-12)^2-4*4*9=144-144=0
x_1=x_2
x=\frac{-b}{2a}=\frac{12}{2*4}=\frac{3}{2}
c)
9x^2 + 1 = 6x
9x^2-6x+1=0
\Delta=\frac{b^2-4ac}{2a}=\frac{(-6)^2-4*9*1}{2*9}=\frac{36-36}{18}=0
\Delta =0, x_1=x_2
x=\frac{-b}{2a}=\frac{6}{2*9}=\frac{2}{6}=\frac{1}{3}
d)
16x^2 + 25 = 40x
16x^2-40x+25=0
\Delta=b^2-4ac=(-40)^2-4*16*25=1600-1600=0
\Delta=0, x_1=x_2
x=\frac{-b}{2a}=\frac40}{2*16}=\frac{40}{32}=1\frac{8}{32}=1\frac{1}{4}=1,25
e)
9x^2 = 12x - 4
9x^2-12x+4=0
\Delta=b^2-4ac=(-12)^2-4*9*4=144-144=0
\Delta=0, x_1=x_2
x=\frac{-b}{2a}=\frac{12}{2*9}=\frac{12}{18}=\frac{2}{3}
f)
25x^2 + 10x + 1 = 0
\Delta=b^2-4ac=10^2-4*25*1=100-100=0
x_1=x_1
x=\frac{-b}{2a}=\frac{-10}{2*25}=\frac{-10}{50}=-\frac{1}{5}=-0,2
Zadanie 4
a)
x^2 + 4x - 5 = 0
\Delta=b^2-4ac=4^2-4*1*(-5)=16+20=36
\sqrt\Delta=\sqrt{36}=6
x_1=\frac{-b-\sqrt\Delta}{2a}=\frac{-4-6}{2*1}=-5
x_2=x_1=\frac{-b+\sqrt\Delta}{2a}=\frac{-4+6}{2*1}=1
b)
-x^2 + 4x +21 = 0
\Delta=b^2-4ac=4^2-4*(-1)*21=16+84=100
\sqrt\Delta=\sqrt{100}=10
x_1=\frac{-b-\sqrt\Delta}{2a}=\frac{-4-10}{2*(-1)}=\frac{-14}{-2}=7
x_2=\frac{-b+\sqrt\Delta}{2a}=\frac{-4+10}{2*(-1)}=\frac{6}{-2}=-3
c)
2x^2 + 5x - 12 = 0
\Delta=b^2-4ac=5^2-4*2*(-12)=25+96=121
\sqrt\Delta=\sqrt{121}=11
x_1=\frac{-b-\sqrt\Delta}{2a}=\frac{-5-11}{2*2}=\frac-16}{4}=-4
x_2=\frac{-b+\sqrt\Delta}{2a}=\frac{-5+11}{2*2}=\frac{6}{4}=\frac{3}{2}=1,5
d)
3x^2 + 7x - 20 = 0
\Delta=b^2-4ac=7^2-4*3*(-20)=49+240=289
\sqrt\Delta=\sqrt{289}=17
x_1=\frac{-b-\sqrt\Delta}{2a}=\frac{-7-17}{2*3}=\frac{-24}{6}=-4
x_2=\frac{-b+\sqrt\Delta}{2a}=\frac{-7+17}{2*3}=\frac{10}{6}=\frac{5}{3}=1\frac{2}{3}
Rozwiązania sprawdzone.