Whakaoti i te (2x-3)(2x+1)=5

$$(2x-3)(2x+1)=5$$

Whakaoti mō x

$$x=-1$$

$$x=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$4x^{2}-4x-3=5$$

Tangohia te 5 mai i ngā taha e rua.

$$4x^{2}-4x-3-5=0$$

Tangohia te 5 i te -3, ka -8.

$$4x^{2}-4x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -4 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4\left(-8\right)}}{2\times 4}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-8\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-4\right)±\sqrt{16-16\left(-8\right)}}{2\times 4}$$

Whakareatia -16 ki te -8.

$$x=\frac{-\left(-4\right)±\sqrt{16+128}}{2\times 4}$$

Tāpiri 16 ki te 128.

$$x=\frac{-\left(-4\right)±\sqrt{144}}{2\times 4}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-\left(-4\right)±12}{2\times 4}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±12}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{4±12}{8}$$

Nā, me whakaoti te whārite x=\frac{4±12}{8} ina he tāpiri te ±. Tāpiri 4 ki te 12.

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

Whakawehe 16 ki te 8.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{4±12}{8} ina he tango te ±. Tango 12 mai i 4.

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

Whakawehe -8 ki te 8.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=2 x=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$4x^{2}-4x-3=5$$

Me tāpiri te 3 ki ngā taha e rua.

$$4x^{2}-4x=5+3$$

Tāpirihia te 5 ki te 3, ka 8.

$$4x^{2}-4x=8$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-4x}{4}=\frac{8}{4}$$

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

$$x^{2}+\left(-\frac{4}{4}\right)x=\frac{8}{4}$$

Whakawehe -4 ki te 4.

$$x^{2}-x=\frac{8}{4}$$

Whakawehe 8 ki te 4.

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

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=2+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=2+\frac{1}{4}$$

Tāpiri 2 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{9}{4}$$

Tauwehea x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{3}{2} x-\frac{1}{2}=-\frac{3}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=2 x=-1$$