Whakaoti i te x^2+6x=-x+30

$$x ^ { 2 } + 6 x = - x + 30$$

Whakaoti mō x

$$x=-10$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te x ki ngā taha e rua.

$$x^{2}+6x+x=30$$

Pahekotia te 6x me x, ka 7x.

$$x^{2}+7x=30$$

Tangohia te 30 mai i ngā taha e rua.

$$x^{2}+7x-30=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+7x-30 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.

$$a+b=7 ab=-30$$

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 7.

$$a=-3 b=10$$

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

$$\left(x-3\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te x+10=0.

$$x=3 x=-10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te x ki ngā taha e rua.

$$x^{2}+6x+x=30$$

Pahekotia te 6x me x, ka 7x.

$$x^{2}+7x=30$$

Tangohia te 30 mai i ngā taha e rua.

$$x^{2}+7x-30=0$$

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.

$$a+b=7 ab=1\left(-30\right)=-30$$

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 7.

$$a=-3 b=10$$

Tuhia anō te x^{2}+7x-30 hei \left(x^{2}-3x\right)+\left(10x-30\right).

$$\left(x^{2}-3x\right)+\left(10x-30\right)$$

Tauwehea te x i te tuatahi me te 10 i te rōpū tuarua.

$$x\left(x-3\right)+10\left(x-3\right)$$

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-3\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te x+10=0.

$$x=3 x=-10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te x ki ngā taha e rua.

$$x^{2}+6x+x=30$$

Pahekotia te 6x me x, ka 7x.

$$x^{2}+7x=30$$

Tangohia te 30 mai i ngā taha e rua.

$$x^{2}+7x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 7 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-30\right)}}{2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-30\right)}}{2}$$

Whakareatia -4 ki te -30.

$$x=\frac{-7±\sqrt{49+120}}{2}$$

Tāpiri 49 ki te 120.

$$x=\frac{-7±\sqrt{169}}{2}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-7±13}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±13}{2} ina he tāpiri te ±. Tāpiri -7 ki te 13.

$$x=\frac{6}{2}$$

Whakawehe 6 ki te 2.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{-7±13}{2} ina he tango te ±. Tango 13 mai i -7.

$$x=-\frac{20}{2}$$

Whakawehe -20 ki te 2.

$$x=-10$$

Kua oti te whārite te whakatau.

$$x=3 x=-10$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te x ki ngā taha e rua.

$$x^{2}+6x+x=30$$

Pahekotia te 6x me x, ka 7x.

$$x^{2}+7x=30$$

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.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=30+\left(\frac{7}{2}\right)^{2}$$

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

$$x^{2}+7x+\frac{49}{4}=30+\frac{49}{4}$$

Tāpiri 30 ki te \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{169}{4}$$

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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{169}{4}$$

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

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

$$x+\frac{7}{2}=\frac{13}{2} x+\frac{7}{2}=-\frac{13}{2}$$

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

$$x=3 x=-10$$