Whakaoti i te 4x^2+7x-17=3x^2+12x-3

$$4 x ^ { 2 } + 7 x - 17 = 3 x ^ { 2 } + 12 x - 3$$

Whakaoti mō x

$$x=-2$$

$$x=7$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 3x^{2} mai i ngā taha e rua.

$$4x^{2}+7x-17-3x^{2}=12x-3$$

Pahekotia te 4x^{2} me -3x^{2}, ka x^{2}.

$$x^{2}+7x-17=12x-3$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+7x-17-12x=-3$$

Pahekotia te 7x me -12x, ka -5x.

$$x^{2}-5x-17=-3$$

Me tāpiri te 3 ki ngā taha e rua.

$$x^{2}-5x-17+3=0$$

Tāpirihia te -17 ki te 3, ka -14.

$$x^{2}-5x-14=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-5x-14 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.

$$a+b=-5 ab=-14$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -14.

$$1,-14 2,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-14=-13 2-7=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-7 b=2$$

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

$$\left(x-7\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+2=0.

$$x=7 x=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3x^{2} mai i ngā taha e rua.

$$4x^{2}+7x-17-3x^{2}=12x-3$$

Pahekotia te 4x^{2} me -3x^{2}, ka x^{2}.

$$x^{2}+7x-17=12x-3$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+7x-17-12x=-3$$

Pahekotia te 7x me -12x, ka -5x.

$$x^{2}-5x-17=-3$$

Me tāpiri te 3 ki ngā taha e rua.

$$x^{2}-5x-17+3=0$$

Tāpirihia te -17 ki te 3, ka -14.

$$x^{2}-5x-14=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-14. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=1\left(-14\right)=-14$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -14.

$$1,-14 2,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-14=-13 2-7=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-7 b=2$$

Tuhia anō te x^{2}-5x-14 hei \left(x^{2}-7x\right)+\left(2x-14\right).

$$\left(x^{2}-7x\right)+\left(2x-14\right)$$

Tauwehea te x i te tuatahi me te 2 i te rōpū tuarua.

$$x\left(x-7\right)+2\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+2=0.

$$x=7 x=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3x^{2} mai i ngā taha e rua.

$$4x^{2}+7x-17-3x^{2}=12x-3$$

Pahekotia te 4x^{2} me -3x^{2}, ka x^{2}.

$$x^{2}+7x-17=12x-3$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+7x-17-12x=-3$$

Pahekotia te 7x me -12x, ka -5x.

$$x^{2}-5x-17=-3$$

Me tāpiri te 3 ki ngā taha e rua.

$$x^{2}-5x-17+3=0$$

Tāpirihia te -17 ki te 3, ka -14.

$$x^{2}-5x-14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -5 mō b, me -14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-14\right)}}{2}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-14\right)}}{2}$$

Whakareatia -4 ki te -14.

$$x=\frac{-\left(-5\right)±\sqrt{25+56}}{2}$$

Tāpiri 25 ki te 56.

$$x=\frac{-\left(-5\right)±\sqrt{81}}{2}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-\left(-5\right)±9}{2}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±9}{2}$$

Nā, me whakaoti te whārite x=\frac{5±9}{2} ina he tāpiri te ±. Tāpiri 5 ki te 9.

$$x=\frac{14}{2}$$

Whakawehe 14 ki te 2.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{5±9}{2} ina he tango te ±. Tango 9 mai i 5.

$$x=-\frac{4}{2}$$

Whakawehe -4 ki te 2.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=7 x=-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3x^{2} mai i ngā taha e rua.

$$4x^{2}+7x-17-3x^{2}=12x-3$$

Pahekotia te 4x^{2} me -3x^{2}, ka x^{2}.

$$x^{2}+7x-17=12x-3$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+7x-17-12x=-3$$

Pahekotia te 7x me -12x, ka -5x.

$$x^{2}-5x-17=-3$$

Me tāpiri te 17 ki ngā taha e rua.

$$x^{2}-5x=-3+17$$

Tāpirihia te -3 ki te 17, ka 14.

$$x^{2}-5x=14$$

Whakawehea te -5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{2}. Nā, tāpiria te pūrua o te -\frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=14+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=14+\frac{25}{4}$$

Tāpiri 14 ki te \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{81}{4}$$

Tauwehea te x^{2}-5x+\frac{25}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{81}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{9}{2} x-\frac{5}{2}=-\frac{9}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=7 x=-2$$