Whakaoti i te x^2+25x=-84

$$x ^ { 2 } + 25 x = - 84$$

Whakaoti mō x

$$x=-21$$

$$x=-4$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 84 ki ngā taha e rua.

$$x^{2}+25x+84=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+25x+84 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.

$$a+b=25 ab=84$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Tātaihia te tapeke mō ia takirua.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

Ko te otinga te takirua ka hoatu i te tapeke 25.

$$a=4 b=21$$

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

$$\left(x+4\right)\left(x+21\right)$$

Hei kimi otinga whārite, me whakaoti te x+4=0 me te x+21=0.

$$x=-4 x=-21$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 84 ki ngā taha e rua.

$$x^{2}+25x+84=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+84. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=25 ab=1\times 84=84$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Tātaihia te tapeke mō ia takirua.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

Ko te otinga te takirua ka hoatu i te tapeke 25.

$$a=4 b=21$$

Tuhia anō te x^{2}+25x+84 hei \left(x^{2}+4x\right)+\left(21x+84\right).

$$\left(x^{2}+4x\right)+\left(21x+84\right)$$

Tauwehea te x i te tuatahi me te 21 i te rōpū tuarua.

$$x\left(x+4\right)+21\left(x+4\right)$$

Whakatauwehea atu te kīanga pātahi x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+4\right)\left(x+21\right)$$

Hei kimi otinga whārite, me whakaoti te x+4=0 me te x+21=0.

$$x=-4 x=-21$$

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.

$$x^{2}+25x=-84$$

Me tāpiri 84 ki ngā taha e rua o te whārite.

$$x^{2}+25x-\left(-84\right)=-84-\left(-84\right)$$

Mā te tango i te -84 i a ia ake anō ka toe ko te 0.

$$x^{2}+25x-\left(-84\right)=0$$

Tango -84 mai i 0.

$$x^{2}+25x+84=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 25 mō b, me 84 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 84}}{2}$$

Pūrua 25.

$$x=\frac{-25±\sqrt{625-4\times 84}}{2}$$

Whakareatia -4 ki te 84.

$$x=\frac{-25±\sqrt{625-336}}{2}$$

Tāpiri 625 ki te -336.

$$x=\frac{-25±\sqrt{289}}{2}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-25±17}{2}$$

Nā, me whakaoti te whārite x=\frac{-25±17}{2} ina he tāpiri te ±. Tāpiri -25 ki te 17.

$$x=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$x=-4$$

Nā, me whakaoti te whārite x=\frac{-25±17}{2} ina he tango te ±. Tango 17 mai i -25.

$$x=-\frac{42}{2}$$

Whakawehe -42 ki te 2.

$$x=-21$$

Kua oti te whārite te whakatau.

$$x=-4 x=-21$$

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.

$$x^{2}+25x=-84$$

Whakawehea te 25, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{25}{2}. Nā, tāpiria te pūrua o te \frac{25}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+25x+\left(\frac{25}{2}\right)^{2}=-84+\left(\frac{25}{2}\right)^{2}$$

Pūruatia \frac{25}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+25x+\frac{625}{4}=-84+\frac{625}{4}$$

Tāpiri -84 ki te \frac{625}{4}.

$$x^{2}+25x+\frac{625}{4}=\frac{289}{4}$$

Tauwehea x^{2}+25x+\frac{625}{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}.

$$\left(x+\frac{25}{2}\right)^{2}=\frac{289}{4}$$

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

$$\sqrt{\left(x+\frac{25}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Whakarūnātia.

$$x+\frac{25}{2}=\frac{17}{2} x+\frac{25}{2}=-\frac{17}{2}$$

Me tango \frac{25}{2} mai i ngā taha e rua o te whārite.

$$x=-4 x=-21$$