Whakaoti i te (x+3)(2x+1)=9

$$( x + 3 ) ( 2 x + 1 ) = 9$$

Whakaoti mō x

$$x=\frac{\sqrt{97}-7}{4}\approx 0.71221445$$

$$x=\frac{-\sqrt{97}-7}{4}\approx -4.21221445$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te 2x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}+7x+3=9$$

Tangohia te 9 mai i ngā taha e rua.

$$2x^{2}+7x+3-9=0$$

Tangohia te 9 i te 3, ka -6.

$$2x^{2}+7x-6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 7 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 2\left(-6\right)}}{2\times 2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 2\left(-6\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-7±\sqrt{49-8\left(-6\right)}}{2\times 2}$$

Whakareatia -8 ki te -6.

$$x=\frac{-7±\sqrt{49+48}}{2\times 2}$$

Tāpiri 49 ki te 48.

$$x=\frac{-7±\sqrt{97}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-7±\sqrt{97}}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{97}}{4} ina he tāpiri te ±. Tāpiri -7 ki te \sqrt{97}.

$$x=\frac{\sqrt{97}-7}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{97}}{4} ina he tango te ±. Tango \sqrt{97} mai i -7.

$$x=\frac{-\sqrt{97}-7}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{97}-7}{4} x=\frac{-\sqrt{97}-7}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te 2x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}+7x+3=9$$

Tangohia te 3 mai i ngā taha e rua.

$$2x^{2}+7x=9-3$$

Tangohia te 3 i te 9, ka 6.

$$2x^{2}+7x=6$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+7x}{2}=\frac{6}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\frac{7}{2}x=\frac{6}{2}$$

Whakawehe 6 ki te 2.

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

Whakawehea te \frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{4}. Nā, tāpiria te pūrua o te \frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=3+\left(\frac{7}{4}\right)^{2}$$

Pūruatia \frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=3+\frac{49}{16}$$

Tāpiri 3 ki te \frac{49}{16}.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{97}{16}$$

Tauwehea x^{2}+\frac{7}{2}x+\frac{49}{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}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{97}{16}$$

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

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{97}{16}}$$

Whakarūnātia.

$$x+\frac{7}{4}=\frac{\sqrt{97}}{4} x+\frac{7}{4}=-\frac{\sqrt{97}}{4}$$

Me tango \frac{7}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{97}-7}{4} x=\frac{-\sqrt{97}-7}{4}$$