Whakaoti i te x^2+16x+64=75-3x^2

$${ x }^{ 2 } +16x+64 = 75-3 { x }^{ 2 }$$

Whakaoti mō x

$$x=\frac{3\sqrt{3}}{2}-2\approx 0.598076211$$

$$x=-\frac{3\sqrt{3}}{2}-2\approx -4.598076211$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 75 mai i ngā taha e rua.

$$x^{2}+16x+64-75=-3x^{2}$$

Tangohia te 75 i te 64, ka -11.

$$x^{2}+16x-11=-3x^{2}$$

Me tāpiri te 3x^{2} ki ngā taha e rua.

$$x^{2}+16x-11+3x^{2}=0$$

Pahekotia te x^{2} me 3x^{2}, ka 4x^{2}.

$$4x^{2}+16x-11=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 16 mō b, me -11 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 4\left(-11\right)}}{2\times 4}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\times 4\left(-11\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-16±\sqrt{256-16\left(-11\right)}}{2\times 4}$$

Whakareatia -16 ki te -11.

$$x=\frac{-16±\sqrt{256+176}}{2\times 4}$$

Tāpiri 256 ki te 176.

$$x=\frac{-16±\sqrt{432}}{2\times 4}$$

Tuhia te pūtakerua o te 432.

$$x=\frac{-16±12\sqrt{3}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-16±12\sqrt{3}}{8}$$

Nā, me whakaoti te whārite x=\frac{-16±12\sqrt{3}}{8} ina he tāpiri te ±. Tāpiri -16 ki te 12\sqrt{3}.

$$x=\frac{12\sqrt{3}-16}{8}$$

Whakawehe -16+12\sqrt{3} ki te 8.

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

Nā, me whakaoti te whārite x=\frac{-16±12\sqrt{3}}{8} ina he tango te ±. Tango 12\sqrt{3} mai i -16.

$$x=\frac{-12\sqrt{3}-16}{8}$$

Whakawehe -16-12\sqrt{3} ki te 8.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 3x^{2} ki ngā taha e rua.

$$x^{2}+16x+64+3x^{2}=75$$

Pahekotia te x^{2} me 3x^{2}, ka 4x^{2}.

$$4x^{2}+16x+64=75$$

Tangohia te 64 mai i ngā taha e rua.

$$4x^{2}+16x=75-64$$

Tangohia te 64 i te 75, ka 11.

$$4x^{2}+16x=11$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+16x}{4}=\frac{11}{4}$$

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

$$x^{2}+\frac{16}{4}x=\frac{11}{4}$$

Whakawehe 16 ki te 4.

$$x^{2}+4x=\frac{11}{4}$$

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

$$x^{2}+4x+2^{2}=\frac{11}{4}+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=\frac{11}{4}+4$$

Tāpiri \frac{11}{4} ki te 4.

$$x^{2}+4x+4=\frac{27}{4}$$

Tauwehea x^{2}+4x+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}.

$$\left(x+2\right)^{2}=\frac{27}{4}$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{\frac{27}{4}}$$

Whakarūnātia.

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

Me tango 2 mai i ngā taha e rua o te whārite.

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