Whakaoti i te 9=49+x^2-13x

$$9 = 49 + x ^ { 2 } - 13 x$$

Whakaoti mō x

$$x=5$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$49+x^{2}-13x=9$$

Tangohia te 9 mai i ngā taha e rua.

$$49+x^{2}-13x-9=0$$

Tangohia te 9 i te 49, ka 40.

$$40+x^{2}-13x=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-13x+40=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-13x+40 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.

$$a+b=-13 ab=40$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Ko te otinga te takirua ka hoatu i te tapeke -13.

$$a=-8 b=-5$$

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

$$\left(x-8\right)\left(x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x-5=0.

$$x=8 x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$49+x^{2}-13x=9$$

Tangohia te 9 mai i ngā taha e rua.

$$49+x^{2}-13x-9=0$$

Tangohia te 9 i te 49, ka 40.

$$40+x^{2}-13x=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-13x+40=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+40. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-13 ab=1\times 40=40$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Ko te otinga te takirua ka hoatu i te tapeke -13.

$$a=-8 b=-5$$

Tuhia anō te x^{2}-13x+40 hei \left(x^{2}-8x\right)+\left(-5x+40\right).

$$\left(x^{2}-8x\right)+\left(-5x+40\right)$$

Tauwehea te x i te tuatahi me te -5 i te rōpū tuarua.

$$x\left(x-8\right)-5\left(x-8\right)$$

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

$$\left(x-8\right)\left(x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x-5=0.

$$x=8 x=5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$49+x^{2}-13x=9$$

Tangohia te 9 mai i ngā taha e rua.

$$49+x^{2}-13x-9=0$$

Tangohia te 9 i te 49, ka 40.

$$40+x^{2}-13x=0$$

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.

$$x^{2}-13x+40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -13 mō b, me 40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 40}}{2}$$

Pūrua -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 40}}{2}$$

Whakareatia -4 ki te 40.

$$x=\frac{-\left(-13\right)±\sqrt{169-160}}{2}$$

Tāpiri 169 ki te -160.

$$x=\frac{-\left(-13\right)±\sqrt{9}}{2}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-\left(-13\right)±3}{2}$$

Ko te tauaro o -13 ko 13.

$$x=\frac{13±3}{2}$$

Nā, me whakaoti te whārite x=\frac{13±3}{2} ina he tāpiri te ±. Tāpiri 13 ki te 3.

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

Whakawehe 16 ki te 2.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{13±3}{2} ina he tango te ±. Tango 3 mai i 13.

$$x=\frac{10}{2}$$

Whakawehe 10 ki te 2.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=8 x=5$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$49+x^{2}-13x=9$$

Tangohia te 49 mai i ngā taha e rua.

$$x^{2}-13x=9-49$$

Tangohia te 49 i te 9, ka -40.

$$x^{2}-13x=-40$$

Whakawehea te -13, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{2}. Nā, tāpiria te pūrua o te -\frac{13}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-40+\left(-\frac{13}{2}\right)^{2}$$

Pūruatia -\frac{13}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-13x+\frac{169}{4}=-40+\frac{169}{4}$$

Tāpiri -40 ki te \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{9}{4}$$

Tauwehea x^{2}-13x+\frac{169}{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}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

$$x-\frac{13}{2}=\frac{3}{2} x-\frac{13}{2}=-\frac{3}{2}$$

Me tāpiri \frac{13}{2} ki ngā taha e rua o te whārite.

$$x=8 x=5$$