Whakaoti i te 4x^2+28x+40=0

$$4 x ^ { 2 } + 28 x + 40 = 0$$

Whakaoti mō x

$$x=-5$$

$$x=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 4.

$$x^{2}+7x+10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=1\times 10=10$$

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 10.

$$1,10 2,5$$

Tātaihia te tapeke mō ia takirua.

$$1+10=11 2+5=7$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=2 b=5$$

Tuhia anō te x^{2}+7x+10 hei \left(x^{2}+2x\right)+\left(5x+10\right).

$$\left(x^{2}+2x\right)+\left(5x+10\right)$$

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

$$x\left(x+2\right)+5\left(x+2\right)$$

Whakatauwehea atu te kīanga pātahi x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+2\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x+2=0 me te x+5=0.

$$x=-2 x=-5$$

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.

$$4x^{2}+28x+40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 28 mō b, me 40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 4\times 40}}{2\times 4}$$

Pūrua 28.

$$x=\frac{-28±\sqrt{784-4\times 4\times 40}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-28±\sqrt{784-16\times 40}}{2\times 4}$$

Whakareatia -16 ki te 40.

$$x=\frac{-28±\sqrt{784-640}}{2\times 4}$$

Tāpiri 784 ki te -640.

$$x=\frac{-28±\sqrt{144}}{2\times 4}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-28±12}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-28±12}{8}$$

Nā, me whakaoti te whārite x=\frac{-28±12}{8} ina he tāpiri te ±. Tāpiri -28 ki te 12.

$$x=-\frac{16}{8}$$

Whakawehe -16 ki te 8.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-28±12}{8} ina he tango te ±. Tango 12 mai i -28.

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

Whakawehe -40 ki te 8.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=-2 x=-5$$

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.

$$4x^{2}+28x+40=0$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$4x^{2}+28x+40-40=-40$$

Mā te tango i te 40 i a ia ake anō ka toe ko te 0.

$$4x^{2}+28x=-40$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+28x}{4}=-\frac{40}{4}$$

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

$$x^{2}+\frac{28}{4}x=-\frac{40}{4}$$

Whakawehe 28 ki te 4.

$$x^{2}+7x=-\frac{40}{4}$$

Whakawehe -40 ki te 4.

$$x^{2}+7x=-10$$

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.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-10+\left(\frac{7}{2}\right)^{2}$$

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

$$x^{2}+7x+\frac{49}{4}=-10+\frac{49}{4}$$

Tāpiri -10 ki te \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{9}{4}$$

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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

$$x+\frac{7}{2}=\frac{3}{2} x+\frac{7}{2}=-\frac{3}{2}$$

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

$$x=-2 x=-5$$