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9k-20-k^{2}+42=0
Whakamahia te āhuatanga tuaritanga hei whakarea te 4-k ki te k-5 ka whakakotahi i ngā kupu rite.
9k+22-k^{2}=0
Tāpirihia te -20 ki te 42, ka 22.
-k^{2}+9k+22=0
Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.
a+b=9 ab=-22=-22
Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -k^{2}+ak+bk+22. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.
-1,22 -2,11
I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -22.
-1+22=21 -2+11=9
Tātaihia te tapeke mō ia takirua.
a=11 b=-2
Ko te otinga te takirua ka hoatu i te tapeke 9.
\left(-k^{2}+11k\right)+\left(-2k+22\right)
Tuhia anō te -k^{2}+9k+22 hei \left(-k^{2}+11k\right)+\left(-2k+22\right).
-k\left(k-11\right)-2\left(k-11\right)
Tauwehea te -k i te tuatahi me te -2 i te rōpū tuarua.
\left(k-11\right)\left(-k-2\right)
Whakatauwehea atu te kīanga pātahi k-11 mā te whakamahi i te āhuatanga tātai tohatoha.
k=11 k=-2
Hei kimi otinga whārite, me whakaoti te k-11=0 me te -k-2=0.
9k-20-k^{2}+42=0
Whakamahia te āhuatanga tuaritanga hei whakarea te 4-k ki te k-5 ka whakakotahi i ngā kupu rite.
9k+22-k^{2}=0
Tāpirihia te -20 ki te 42, ka 22.
-k^{2}+9k+22=0
Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.
k=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\times 22}}{2\left(-1\right)}
Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 9 mō b, me 22 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
k=\frac{-9±\sqrt{81-4\left(-1\right)\times 22}}{2\left(-1\right)}
Pūrua 9.
k=\frac{-9±\sqrt{81+4\times 22}}{2\left(-1\right)}
Whakareatia -4 ki te -1.
k=\frac{-9±\sqrt{81+88}}{2\left(-1\right)}
Whakareatia 4 ki te 22.
k=\frac{-9±\sqrt{169}}{2\left(-1\right)}
Tāpiri 81 ki te 88.
k=\frac{-9±13}{2\left(-1\right)}
Tuhia te pūtakerua o te 169.
k=\frac{-9±13}{-2}
Whakareatia 2 ki te -1.
k=\frac{4}{-2}
Nā, me whakaoti te whārite k=\frac{-9±13}{-2} ina he tāpiri te ±. Tāpiri -9 ki te 13.
k=-2
Whakawehe 4 ki te -2.
k=-\frac{22}{-2}
Nā, me whakaoti te whārite k=\frac{-9±13}{-2} ina he tango te ±. Tango 13 mai i -9.
k=11
Whakawehe -22 ki te -2.
k=-2 k=11
Kua oti te whārite te whakatau.
9k-20-k^{2}+42=0
Whakamahia te āhuatanga tuaritanga hei whakarea te 4-k ki te k-5 ka whakakotahi i ngā kupu rite.
9k+22-k^{2}=0
Tāpirihia te -20 ki te 42, ka 22.
9k-k^{2}=-22
Tangohia te 22 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.
-k^{2}+9k=-22
Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.
\frac{-k^{2}+9k}{-1}=-\frac{22}{-1}
Whakawehea ngā taha e rua ki te -1.
k^{2}+\frac{9}{-1}k=-\frac{22}{-1}
Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.
k^{2}-9k=-\frac{22}{-1}
Whakawehe 9 ki te -1.
k^{2}-9k=22
Whakawehe -22 ki te -1.
k^{2}-9k+\left(-\frac{9}{2}\right)^{2}=22+\left(-\frac{9}{2}\right)^{2}
Whakawehea te -9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{2}. Nā, tāpiria te pūrua o te -\frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.
k^{2}-9k+\frac{81}{4}=22+\frac{81}{4}
Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.
k^{2}-9k+\frac{81}{4}=\frac{169}{4}
Tāpiri 22 ki te \frac{81}{4}.
\left(k-\frac{9}{2}\right)^{2}=\frac{169}{4}
Tauwehea k^{2}-9k+\frac{81}{4}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(k-\frac{9}{2}\right)^{2}}=\sqrt{\frac{169}{4}}
Tuhia te pūtakerua o ngā taha e rua o te whārite.
k-\frac{9}{2}=\frac{13}{2} k-\frac{9}{2}=-\frac{13}{2}
Whakarūnātia.
k=11 k=-2
Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.