a)
Miejsca zerowe funkcji f:
[tex]f(x) = x^{2}+4x-5\\\\f(x) = 0\\\\x^{2}+4x-5 = 0\\\\a = 1, \ b = 4, \ c = -5\\\\\Delta = b^{2}-4a=4^{2}-4\cdot1\cdot(-5) = 16+20 = 36\\\\\Sqrt{\Delta} = \sqrt{36} = 6\\\\x_1 =\frac{-b-\sqrt{\Delta}}{2a} = \frac{-4-6}{2\cdot1} = \frac{-10}{2} = -5\\\\x_2 = \frac{-b+\sqrt{\Delta}}{2a} =\frac{-4+6}{2} = \frac{2}{2} = 1[/tex]
Postać iloczynowa:
[tex]f(x) = a(x-x_1)(x-x_2)\\\\\boxed{f(x) = (x+5)(x-1)} \ - \ potac \ iloczynowa[/tex]
b)
Wspólrzędne wierzchołka paraboli:
[tex]W = (p, q)\\\\p = \frac{x_1+x_2}{2} = \frac{-5+1}{2} = \frac{-4}{2} = -2\\\\q = \frac{-\Delta}{4a} = \frac{-36}{4\cdot1} = \frac{-36}{4}= -9\\\\\boxed{W = (-2, -9)}[/tex]
Postać kanoniczna:
[tex]f(x) = a(x-p)^{2}+q\\\\\boxed{f(x) = (x+2)^{2}-9} \ - \ postac \ kanoniczna[/tex]
