Whakaoti i te 385=(2x+2)*(2x+3)

$$385=(2x+2) \times (2x+3)$$

Whakaoti mō x

$$x = \frac{\sqrt{1541} - 5}{4} \approx 8.563893213$$

$$x=\frac{-\sqrt{1541}-5}{4}\approx -11.063893213$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+2 ki te 2x+3 ka whakakotahi i ngā kupu rite.

$$385=4x^{2}+10x+6$$

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

$$4x^{2}+10x+6=385$$

Tangohia te 385 mai i ngā taha e rua.

$$4x^{2}+10x+6-385=0$$

Tangohia te 385 i te 6, ka -379.

$$4x^{2}+10x-379=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 10 mō b, me -379 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 4\left(-379\right)}}{2\times 4}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 4\left(-379\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-10±\sqrt{100-16\left(-379\right)}}{2\times 4}$$

Whakareatia -16 ki te -379.

$$x=\frac{-10±\sqrt{100+6064}}{2\times 4}$$

Tāpiri 100 ki te 6064.

$$x=\frac{-10±\sqrt{6164}}{2\times 4}$$

Tuhia te pūtakerua o te 6164.

$$x=\frac{-10±2\sqrt{1541}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-10±2\sqrt{1541}}{8}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{1541}}{8} ina he tāpiri te ±. Tāpiri -10 ki te 2\sqrt{1541}.

$$x=\frac{2\sqrt{1541}-10}{8}$$

Whakawehe -10+2\sqrt{1541} ki te 8.

$$x=\frac{\sqrt{1541}-5}{4}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{1541}}{8} ina he tango te ±. Tango 2\sqrt{1541} mai i -10.

$$x=\frac{-2\sqrt{1541}-10}{8}$$

Whakawehe -10-2\sqrt{1541} ki te 8.

$$x=\frac{-\sqrt{1541}-5}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{1541}-5}{4} x=\frac{-\sqrt{1541}-5}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+2 ki te 2x+3 ka whakakotahi i ngā kupu rite.

$$385=4x^{2}+10x+6$$

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

$$4x^{2}+10x+6=385$$

Tangohia te 6 mai i ngā taha e rua.

$$4x^{2}+10x=385-6$$

Tangohia te 6 i te 385, ka 379.

$$4x^{2}+10x=379$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+10x}{4}=\frac{379}{4}$$

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

$$x^{2}+\frac{10}{4}x=\frac{379}{4}$$

Whakahekea te hautanga \frac{10}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{5}{2}x=\frac{379}{4}$$

Whakawehea te \frac{5}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{4}. Nā, tāpiria te pūrua o te \frac{5}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=\frac{379}{4}+\left(\frac{5}{4}\right)^{2}$$

Pūruatia \frac{5}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{379}{4}+\frac{25}{16}$$

Tāpiri \frac{379}{4} ki te \frac{25}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{1541}{16}$$

Tauwehea x^{2}+\frac{5}{2}x+\frac{25}{16}. 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}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{1541}{16}$$

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

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{1541}{16}}$$

Whakarūnātia.

$$x+\frac{5}{4}=\frac{\sqrt{1541}}{4} x+\frac{5}{4}=-\frac{\sqrt{1541}}{4}$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{1541}-5}{4} x=\frac{-\sqrt{1541}-5}{4}$$