Whakaoti i te x^2+7x-78=42

$$x ^ { 2 } + 7 x - 78 = 42$$

Whakaoti mō x

$$x=-15$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 42 mai i ngā taha e rua.

$$x^{2}+7x-78-42=0$$

Tangohia te 42 i te -78, ka -120.

$$x^{2}+7x-120=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+7x-120 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=-120$$

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 -120.

$$-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12$$

Tātaihia te tapeke mō ia takirua.

$$-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=-8 b=15$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-8\right)\left(x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+15=0.

$$x=8 x=-15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 42 mai i ngā taha e rua.

$$x^{2}+7x-78-42=0$$

Tangohia te 42 i te -78, ka -120.

$$x^{2}+7x-120=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-120. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=1\left(-120\right)=-120$$

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 -120.

$$-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12$$

Tātaihia te tapeke mō ia takirua.

$$-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=-8 b=15$$

Tuhia anō te x^{2}+7x-120 hei \left(x^{2}-8x\right)+\left(15x-120\right).

$$\left(x^{2}-8x\right)+\left(15x-120\right)$$

Tauwehea te x i te tuatahi me te 15 i te rōpū tuarua.

$$x\left(x-8\right)+15\left(x-8\right)$$

Whakatauwehea atu te kīanga pātahi x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-8\right)\left(x+15\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+15=0.

$$x=8 x=-15$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}+7x-78=42$$

Me tango 42 mai i ngā taha e rua o te whārite.

$$x^{2}+7x-78-42=42-42$$

Mā te tango i te 42 i a ia ake anō ka toe ko te 0.

$$x^{2}+7x-78-42=0$$

Tango 42 mai i -78.

$$x^{2}+7x-120=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 7 mō b, me -120 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-120\right)}}{2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-120\right)}}{2}$$

Whakareatia -4 ki te -120.

$$x=\frac{-7±\sqrt{49+480}}{2}$$

Tāpiri 49 ki te 480.

$$x=\frac{-7±\sqrt{529}}{2}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-7±23}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±23}{2} ina he tāpiri te ±. Tāpiri -7 ki te 23.

$$x=\frac{16}{2}$$

Whakawehe 16 ki te 2.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{-7±23}{2} ina he tango te ±. Tango 23 mai i -7.

$$x=-\frac{30}{2}$$

Whakawehe -30 ki te 2.

$$x=-15$$

Kua oti te whārite te whakatau.

$$x=8 x=-15$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x^{2}+7x-78=42$$

Me tāpiri 78 ki ngā taha e rua o te whārite.

$$x^{2}+7x-78-\left(-78\right)=42-\left(-78\right)$$

Mā te tango i te -78 i a ia ake anō ka toe ko te 0.

$$x^{2}+7x=42-\left(-78\right)$$

Tango -78 mai i 42.

$$x^{2}+7x=120$$

Whakawehea te 7, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{2}. Nā, tāpiria te pūrua o te \frac{7}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=120+\left(\frac{7}{2}\right)^{2}$$

Pūruatia \frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+7x+\frac{49}{4}=120+\frac{49}{4}$$

Tāpiri 120 ki te \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{529}{4}$$

Tauwehea x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{529}{4}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{529}{4}}$$

Whakarūnātia.

$$x+\frac{7}{2}=\frac{23}{2} x+\frac{7}{2}=-\frac{23}{2}$$

Me tango \frac{7}{2} mai i ngā taha e rua o te whārite.

$$x=8 x=-15$$