Whakaoti i te 4x^2+4x=120

$$4 { x }^{ 2 } +4x=120$$

Whakaoti mō x

$$x=-6$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 120 mai i ngā taha e rua.

$$4x^{2}+4x-120=0$$

Whakawehea ngā taha e rua ki te 4.

$$x^{2}+x-30=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=1\left(-30\right)=-30$$

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 -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-5 b=6$$

Tuhia anō te x^{2}+x-30 hei \left(x^{2}-5x\right)+\left(6x-30\right).

$$\left(x^{2}-5x\right)+\left(6x-30\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x-5\right)+6\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+6=0.

$$x=5 x=-6$$

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.

$$4x^{2}+4x=120$$

Me tango 120 mai i ngā taha e rua o te whārite.

$$4x^{2}+4x-120=120-120$$

Mā te tango i te 120 i a ia ake anō ka toe ko te 0.

$$4x^{2}+4x-120=0$$

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

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

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 4\left(-120\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-4±\sqrt{16-16\left(-120\right)}}{2\times 4}$$

Whakareatia -16 ki te -120.

$$x=\frac{-4±\sqrt{16+1920}}{2\times 4}$$

Tāpiri 16 ki te 1920.

$$x=\frac{-4±\sqrt{1936}}{2\times 4}$$

Tuhia te pūtakerua o te 1936.

$$x=\frac{-4±44}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-4±44}{8}$$

Nā, me whakaoti te whārite x=\frac{-4±44}{8} ina he tāpiri te ±. Tāpiri -4 ki te 44.

$$x=\frac{40}{8}$$

Whakawehe 40 ki te 8.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{-4±44}{8} ina he tango te ±. Tango 44 mai i -4.

$$x=-\frac{48}{8}$$

Whakawehe -48 ki te 8.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=5 x=-6$$

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.

$$4x^{2}+4x=120$$

Whakawehea ngā taha e rua ki te 4.

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

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

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

Whakawehe 4 ki te 4.

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

Whakawehe 120 ki te 4.

$$x^{2}+x=30$$

Whakawehea te 1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{2}. Nā, tāpiria te pūrua o te \frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=30+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=30+\frac{1}{4}$$

Tāpiri 30 ki te \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{121}{4}$$

Tauwehea x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{11}{2} x+\frac{1}{2}=-\frac{11}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=5 x=-6$$